Plane Wave Discontinuous Galerkin Methods: Exponential Convergence of the $hp$-Version

Abstract

We consider the two-dimensional Helmholtz equation with constant coefficients on a domain with piecewise analytic boundary, modelling the scattering of acoustic waves at a sound-soft obstacle. Our discretisation relies on the Trefftz-discontinuous Galerkin approach with plane wave basis functions on meshes with very general element shapes, geometrically graded towards domain corners. We prove exponential convergence of the discrete solution in terms of number of unknowns.

1 Introduction

This article is concerned with a particular type of Trefftz method for 2D scalar wave scattering problems in the frequency domain, modelled by means of the linear Helmholtz equation with constant coefficients. In general, Trefftz methods try to incorporate information about the exact solution into local approximation spaces by requiring that they are contained in the kernel of the governing differential operator. This policy looks particularly attractive for wave propagation, which usually involves oscillatory solutions.

It is not straightforward to marry the Trefftz idea with classical conforming finite element Galerkin discretisations, cf. the partition of unity method [3, 25]. Conversely, discontinuous Galerkin (DG) methods, which do not impose any interelement continuity on the trial functions, offer a very convenient framework for the implementation of Trefftz methods.

For wave propagation problems in homogeneous media, natural Trefftz functions are plane waves, which give rise to plane wave discontinuous Galerkin (PWDG) methods. Their oldest representative is the so-called ultra-weak variational formulation (UWVF), proposed in [7]. It was not recognised as a PWDG method in the beginning, and a comprehensive convergence theory remained elusive for quite some time. Finally, in [6, 11, 13], the UWVF was recast as a DG method, thus paving the way for using the powerful arsenal of DG analysis.

The first fruit was harvested in [13] in the form of a complete convergence analysis of the $h$-version of PWDG. The $h$-version was also tackled independently in [6], based on tools from [35]. It turned out that these tools could also be harnessed to deal with the $p$-version, and this was done in [17]. Algebraic convergence in $p$ could be established, though confined to “quasi-uniform” meshes. Of course, here, instead of designating the polynomial degree, $p$ should be read as the number of plane waves used for local approximation. Later, in [19], the $p$-convergence theory was extended to cover locally refined meshes.

Based on the techniques from [19], in this article, we pursue the ultimate goal of establishing exponential convergence (with respect to the number of degrees of freedom) of PWDG solutions, when the trial spaces are built following a policy borrowed from standard $hp$-finite element methods. Assuming domains and data with sufficient regularity, the idea is to use large mesh cells equipped with many plane waves where the solution is smooth, whereas small cells are employed to resolve singularities of the solution at corners of the boundary. This kind of $hp$ approximation with polynomials has seen an amazing development starting from the work of Babuška [2, 15]; see [38] for a comprehensive exposition. It has also been adapted to polynomial DG methods by several authors, see, for instance [21, 36, 37, 41]. Applications to scalar wave propagation are reported in [10, 28, 29].

Results on the approximation of Helmholtz solutions by plane waves are pivotal for our estimates. In this direction, major progress has been achieved in [31, 32]. These works make use of Vekua’s theory and, thus, could exploit known results about the approximation of harmonic functions by harmonic polynomials. Recently, results in this direction targeting harmonic functions that can be extended analytically were obtained in [20], generalising earlier work by Melenk [25]. A proof of exponential convergence of the $hp$-version of (polynomial) Trefftz-DG method for the Laplace problem was included.

The main result of this work (Theorem 6.5, Sect. 6) is a proof that the $L^{2}$-norm of the discretisation error of a special PWDG method on very general, geometrically graded meshes converges exponentially in the square root of the number of degrees of freedom. This is the first such result for a numerical method based on plane waves. For the proof, we had to refine the duality arguments of [19], see Sect. 4, and combine them with novel $L^{\infty }$-approximation estimates for plane waves given in Sect. 5. The reason of the restriction to two space dimensions is that the approximation estimates for harmonic functions we rely on (see Proposition 5.1) were derived in [20] using complex analysis arguments and thus are proved in 2D only. We find that the error is bounded by a negative exponential of the square root of the total number of degrees of freedom employed, while typical polynomial $hp$-schemes in two dimensions only deliver exponential convergence in the cubic root of the same parameter, e.g. see [2], Theorem 5.3]. The results of our analysis hold true also when circular waves are used instead of plane waves. For simplicity, we assume that the computational domain is the set difference between two star-shaped domains with common centre; however, this geometrical setting can easily be generalised, see Remark 2.2.

At this point, we emphasise that our focus is on numerical approximation theory. We deliberately ignore the key challenge of ill-conditioning of linear systems arising from PWDG approaches, cf. [22, 23]. We even acknowledge that an implementation of the method investigated below may severely be affected by numerical instability, see Remark 6.8.

2 Scattering Boundary Value Problem

As in [19], Sect. 2], let $\Omega _D\subset \mathbb {R}^2$ be a bounded, Lipschitz domain occupied by a sound-soft material, which we assume to be star-shaped with respect to the origin ${\mathbf {0}}$. We denote by $\Gamma _D:=\partial \Omega _{D}$ its boundary. We introduce another bounded Lipschitz domain $\Omega _R$ with boundary $\Gamma _R$ such that $\overline{\Omega }_D\subset \Omega _R$, and ${{\mathrm{dist}}}(\Gamma _D,\Gamma _R)>0$.¹ We set $\Omega :=\Omega _R\setminus \overline{\Omega }_D$, and we assume $\partial \Omega$ to be piecewise analytic. It may have finitely many corners ${{\mathbf {c}}}_\nu$, $1\le \nu \le n_c$, which we collect in the set ${\mathcal {C}}:= \left\{ {{\mathbf {c}}}_{\nu }\right\} _{\nu =1}^{n_c}$. By scaling, we can always achieve $\underline{{\text {diam}}(\Omega )=1}$, which we take for granted throughout the remainder of the article.

We focus on the following boundary value problem (BVP) for the Helmholtz equation:

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u-k^2 u=0 &{}\;\text {in}\;\Omega ,\\ u=0 &{} \;\text {on}\;\Gamma _D,\\ \nabla u\cdot {{\mathbf {n}}}+ik\vartheta u =g_R &{}\;\text {on}\;\Gamma _R, \end{array}\right. } \end{aligned}$$

(1)

with $g_R\in L^2(\Gamma _R)$, real wavenumber $k$, and $\vartheta \in \mathbb {R}$ a non-dimensional, nonzero parameter. Since our focus is on true wave propagation problems, in the sequel, we assume $k>1$. In (1), we have written ${{\mathbf {n}}}$ for the outward-pointing unit normal vector field on ${\partial \Omega }$.

2.1 Stability and Sobolev Regularity

We denote by $\left\| \cdot \right\| _{0,D}$ the $L^2(D)$-norm and by $\left| \cdot \right| _{\ell ,D}$ the $H^\ell (D)$-Sobolev seminorm, $\ell \in {\mathbb {N}}_0$ (${\mathbb {N}}_0=\{0,1,2,\ldots \}$), where $D$ is a Lipschitz domain. For positive non-integer values of $s$, we consider the $H^s(D)$-seminorm as defined by the Sobolev–Slobodeckij integral (see e.g. [34], Page 43]). On a Lipschitz manifold $D$, we use only the $L^2(D)$-norm and the $H^s(D)$-seminorm for $0<s<1$. It is convenient to make use of the following $k$-weighted Sobolev norms:

$$\begin{aligned} \left\| v\right\| _{\ell ,k,D}^2:=\sum _{j=0}^\ell k^{2(\ell -j)} \left| u\right| _{j,D}^2\qquad \forall v\in H^\ell (D),\; \ell \in {\mathbb {N}}. \end{aligned}$$

We assume that $\Omega _R$ is star-shaped with respect to the ball² $B_{\gamma _R}$, for some $\gamma _R>0$. Next, we sharpen Theorems 2.1, 2.2 and 2.3 of [19] (see also [16], Propositions 3.3 and 3.4]) and obtain the following stability and elliptic regularity result.

Proposition 2.1

Let $u$ be the solution of the inhomogeneous boundary value problem

$$\begin{aligned} -\Delta u-k^2 u&=f \qquad \;\text {in}\;\Omega , \end{aligned}$$

(2)

$$\begin{aligned} u&= 0 \qquad \;\text {on}\;\Gamma _D, \end{aligned}$$

(3)

$$\begin{aligned} \nabla u\cdot {{\mathbf {n}}}\pm ik\vartheta u&= g_R \quad \ \;\text {on}\;\Gamma _R. \end{aligned}$$

(4)

If $f\in L^{2}(\Omega )$ and $g_R\in L^2(\Gamma _R)$, the weak formulation of (2)–(4) is well posed in $H^1(\Omega )$. Moreover, if $g_R\in H^{r}(\Gamma _R)$ for a given $0<r<1/2$, then there exists $s_{\Omega }>0$ depending only on (the corners of) $\Omega$, such that $u\in H^{1+t}(\Omega )$ for every $t$ satisfying

$$\begin{aligned} {0\le t < \frac{1}{2}+ s_\Omega , \qquad t \le r + \frac{1}{2}}, \end{aligned}$$

(5)

and the following bounds hold:

$$\begin{aligned} \left\| u\right\| _{1,k,\Omega } + k \left\| u\right\| _{0,\Gamma _{R}}&\le C\Big (\left\| f\right\| _{0,\Omega }+\left\| g_R\right\| _{0,\Gamma _R}\Big ), \end{aligned}$$

(6)

$$\begin{aligned} \left| \nabla u\right| _{t,\Omega }&\le C(1+ k^{t})\Big (\left\| f\right\| _{0,\Omega } + \left\| g_R\right\| _{0,\Gamma _R}\Big )+C \left| g_R\right| _{t-\frac{1}{2},\Gamma _R}, \end{aligned}$$

(7)

where the constants depend only on $t$, $\gamma _R$ and $\vartheta$, but are independent of $k$, $f$, $g_R$ and $u$.

Proof

For the estimate (6), we refer the reader to [12], Theorem 2.18], [33], Sect. 4] and, in particular [33], Remark 4.7].

To prove (7), we first consider $\Omega _{D}=\emptyset$. In this case, we appeal to [8], Corollary 23.5], to [14], Theorem 2.4.3 and Remark 2.4.5], and interpolation between $\widetilde{H}^{-1}(\Omega )$ and $\widetilde{H}^{-\frac{1}{2}+\sigma }(\Omega )$, and $H^{-\frac{1}{2}}(\Gamma _{R})$ and $H^{\sigma }(\Gamma _{R})$ for some $\sigma >0$, to argue that we can find $s_{\Omega }>0$ depending only on $\Omega$ such that $u\in H^{1+t}(\Omega )$ for all $t$ satisfying (5),and

$$\begin{aligned} \left| \nabla u\right| _{t,\Omega } \le C \bigl (\left\| \Delta u\right\| _{\widetilde{H}^{t-1}(\Omega )} + \left\| \nabla u\cdot {{\mathbf {n}}}\right\| _{t-\frac{1}{2},\Gamma _{R}}\bigr ), \end{aligned}$$

(8)

where $\Delta :H^{1}(\Omega )\rightarrow \widetilde{H}^{-1}(\Omega )$ is the (Neumann-)Laplacian in weak form. In (8) and throughout the remainder of the proof, all constants may depend only on $t$, $\Omega _{R}$, $\Omega _{D}$ and $\vartheta$.

We use the impedance boundary condition (4) to replace the normal derivative in (8) and find

$$\begin{aligned} \left\| \nabla u\cdot {{\mathbf {n}}}\right\| _{t-\frac{1}{2},\Gamma _{R}} \le C\bigl ( \left\| g_{R}\right\| _{t-\frac{1}{2},\Gamma _{R}} + k\left\| u\right\| _{t-\frac{1}{2},\Gamma _{R}}\bigr ). \end{aligned}$$

Next, we distinguish two cases: (i) If $t\le \frac{1}{2}$, that is $t-\frac{1}{2}\le 0$, then from (6)

$$\begin{aligned} \left\| \nabla u\cdot {{\mathbf {n}}}\right\| _{t-\frac{1}{2},\Gamma _{R}} \overset{(4)}{\le } C \bigl (\left\| g_{R}\right\| _{0,\Gamma _{R}} + k \left\| u\right\| _{0,\Gamma _{R}}\bigr ) \le C \bigl (\left\| g_{R}\right\| _{0,\Gamma _{R}} + \left\| f\right\| _{0,\Omega }\bigr ). \end{aligned}$$

(9)

(ii) If $t>\frac{1}{2}$, we resort to an interpolation estimate in the Sobolev scale [24], Lemma B.1] and find ($0< t-\frac{1}{2}\le \frac{1}{2}$)

$$\begin{aligned} k\left\| u\right\| _{t-\frac{1}{2},\Gamma _{R}} \le Ck \left\| u\right\| _{\frac{1}{2},\Gamma _{R}}^{2t-1} \left\| u\right\| _{0,\Gamma _{R}}^{2(1-t)}. \end{aligned}$$

We bound $\left\| u\right\| _{\frac{1}{2},\Gamma _{R}}$ by the trace theorem [24], Theorem 3.37], and $\left\| u\right\| _{0,\Gamma _{R}}$ by a multiplicative trace estimate [5], Theorem 1.6.6] and get

$$\begin{aligned} k\left\| u\right\| _{t-\frac{1}{2},\Gamma _{R}}&\le Ck \bigl ( \left\| u\right\| _{0,\Omega } + \left\| \nabla u\right\| _{0,\Omega } \bigr )^{2t-1} \left\| u\right\| _{0,\Omega }^{1-t}\bigl (\left\| u\right\| _{0,\Omega }+\left\| \nabla u\right\| _{0,\Omega }\bigr )^{1-t} \nonumber \\&\le Ck^{t} \bigl (\left\| f\right\| _{0,\Omega }+\left\| g_{R}\right\| _{0,\Gamma _{R}}), \end{aligned}$$

(10)

where we used (6) in the last step.

Another interpolation estimate in the dual Sobolev scale, see [24], Theorems 3.30 and B.9], yields

$$\begin{aligned} \left\| \Delta u\right\| _{\widetilde{H}^{t-1}(\Omega )}&\le C \left\| \Delta u\right\| _{0,\Omega }^{t}\left\| \Delta u\right\| _{\widetilde{H}^{-1}(\Omega )}^{1-t} \overset{(2)}{\le } C \bigl (\left\| f\right\| _{0,\Omega } + k^{2}\left\| u\right\| _{0,\Omega })^{t} \left\| \nabla u\right\| _{0,\Omega }^{1-t}\\&\overset{(6)}{\le } C \bigl (\left\| f\right\| _{0,\Omega } + k( \left\| f\right\| _{0,\Omega }+\left\| g_{R}\right\| _{0,\Gamma _{R}} )\bigr )^{t} \bigl (\left\| f\right\| _{0,\Omega }+\left\| g_{R}\right\| _{0,\Gamma _{R}} \bigr )^{1-t} \\&\le C (1+k^{t})\bigl ( \left\| f\right\| _{0,\Omega }+\left\| g_{R}\right\| _{0,\Gamma _{R}} \bigr ). \end{aligned}$$

Combining this with (8) and using (9) together with (10), we arrive at (7) in the situation $\Omega _{D}=\emptyset$.

To extend the estimate to the presence of a scatterer occupying $\Omega _{D}\not =\emptyset$, we can continue exactly as in the second part of the proof of [19], Theorem 2.3]. $\square$

Remark 2.2

(Non-star-shaped domains) In the case of an interior impedance problem (i.e. where $\Omega =\Omega _R$ and $\Omega _D=\emptyset$), $k$-explicit stability bounds have been proved in [9], Theorem 2.4] and improved in their $k$-dependence in [39], Theorem 1.6], without assuming $\Omega$ to be star-shaped. If the scatterer $\Omega _D$ is Lipschitz but trapping, thus not star-shaped, the constants in the stability bounds may grow exponentially in $k$, as shown in [4], Theorem 2.8] (note in particular Eq. (2.22) in [4], and that the functions $v_m$ in the proof of [4], Theorem 2.8] are compactly supported thus satisfy the boundary value problem (2)–(4) in a suitable $\Omega _R$). The Sobolev regularity of $u$ is not affected as long as $\Omega$ is Lipschitz.

2.2 Analytic Regularity

In this section, we state an analytic regularity result for the solution $u$ to problem (1). This result is derived within the setting of [26], Chapters 4 and 5], which extends the theory of Babuška and Guo [2] to the case of general elliptic equations with a perturbation parameter. We essentially combine the $L^2$-estimates of the derivatives of $u$ given in [26], Chapter 5] with the $L^\infty$-estimates of [2], Theorem 2.2].

To translate our problem into the notation of [26], as in [29], proof of Lemma 4.13], we set

$$\begin{aligned} A({{\mathbf {x}}})=1,\qquad f({{\mathbf {x}}})=0\ \ (C_f=0),\qquad b({{\mathbf {x}}})=0\ \ (C_b=0),\qquad c({{\mathbf {x}}})=1\ \ (C_c=1); \end{aligned}$$

the perturbation parameter is

$$\begin{aligned} \varepsilon =\frac{1}{ik}, \end{aligned}$$

and therefore the length scale is ${\mathcal {E}}=\displaystyle {\frac{1}{k+1}}$, and $\displaystyle {\frac{{\mathcal {E}}}{\left| \varepsilon \right| }}\le 1$. Comparing the expression of the Robin boundary condition, we also set

$$\begin{aligned} G_1=\frac{1}{ik}g_R\ \ \Big (C_{G_1}=\frac{1}{k}\left\| g_R\right\| _{H^{1/2}(\Gamma _R)}\Big ),\qquad G_2=-\vartheta \ \ \left( C_{G_2}=\left| \vartheta \right| \right) . \end{aligned}$$

Recalling that $n_c$ is the number of corner points of $\partial \Omega$, given $\underline{\beta }\in [0,1)^{n_c}$ and $\ell \in {\mathbb {N}}_0$, let ${\mathcal {B}}_{\underline{\beta },{\mathcal {E}}}^\ell (\Omega )$ be the countably normed spaces defined in [26], Chapter 4] (see also [29], Sect. 1.1]), with weights given by

$$\begin{aligned} \widehat{\Phi }_{p,\underline{\beta },{\mathcal {E}}}({{\mathbf {x}}})= \prod _{\nu =1}^{n_c}\Phi _{p,\beta _\nu ,{\mathcal {E}}}({{\mathbf {x}}}-{{\mathbf {c}}}_\nu )\qquad \forall p\in {\mathbb {N}}_0, \end{aligned}$$

where

$$\begin{aligned} \Phi _{p,\beta ,{\mathcal {E}}}({{\mathbf {x}}})=\min \left\{ 1,\frac{\left| {{\mathbf {x}}}\right| }{\min \{1,{\mathcal {E}}(\left| p\right| +1)\}} \right\} ^{p+\beta }. \end{aligned}$$

We set $\widehat{\Phi }({{\mathbf {x}}}):=\widehat{\Phi }_{1,\underline{0},1}({{\mathbf {x}}}) {=\prod _{\nu =1}^{n_c}|{{\mathbf {x}}}-{{\mathbf {c}}}_\nu |}$, which, obviously, is independent of $k$.

Theorem 2.3

There exists a weight vector $\underline{\beta }\in (0,1)^{n_c}$ such that, if $g_R\in {\mathcal {B}}_{\underline{\beta },{\mathcal {E}}}^1(\Gamma _R)$, the solution $u$ to problem (1) belongs to ${\mathcal {B}}_{\underline{\beta },{\mathcal {E}}}^2(\Omega )$. Moreover, there exists a constant $\gamma >0$ independent of $k$ such that $u$ admits a real analytic continuation to the set

$$\begin{aligned} {\mathcal {N}}(u):=\bigcup _{{{\mathbf {x}}}_0 \in \overline{\Omega }\setminus \bigcup _{\nu =1}^{n_c} {{\mathbf {c}}}_{\nu }} \left\{ {{\mathbf {x}}}\in \mathbb {R}^2:\ |{{\mathbf {x}}}-{{\mathbf {x}}}_0|< \frac{\widehat{\Phi }({{\mathbf {x}}}_0)}{4e\,\gamma }\right\} \subset \mathbb {R}^2. \end{aligned}$$

(11)

Proof

(Sketch) Within the general setting of [26], Chapter 5], since $\Gamma _D\cap \Gamma _R=\emptyset$ (thus Dirichlet and Robin boundaries do not affect one another), Theorem 5.3.10 and Proposition 5.4.5 (see also Remark 5.3.11 and 5.4.6) of [26] can be applied, and taking into account (6), one can conclude that $ku\in {\mathcal {B}}_{\underline{\beta },{\mathcal {E}}}^2(\Omega )$ for some $\underline{\beta }\in (0,1)^{n_c}$. In particular, denoting by $\nabla ^\ell$ the derivatives of order $\ell$ (more precisely, $\left| \nabla ^\ell u({{\mathbf {x}}})\right| ^2=\sum _{{\varvec{\alpha }}\in {\mathbb {N}}_0^2,\left| {\varvec{\alpha }}\right| =\ell } \frac{\ell !}{{\varvec{\alpha }}!}\left| D^{\varvec{\alpha }}u({{\mathbf {x}}})\right| ^2$),

$$\begin{aligned} \left\| \widehat{\Phi }_{p,\underline{\beta },{\mathcal {E}}}\nabla ^{p+2}u\right\| _{0,\Omega } \le C\,\big (\gamma \,\max \{p+2,k\}\big )^{p+2}k^{-1}\qquad \forall p\in {\mathbb {N}}_0, \end{aligned}$$

in addition to

$$\begin{aligned} \left\| u\right\| _{0,\Omega }\le Ck^{-1}, \qquad \left\| \nabla u\right\| _{0,\Omega }\le C \end{aligned}$$

(see also [29], Lemma 4.13]); here and in the remainder of this proof, $C$ and $\gamma$ are positive constants independent of $k$ ($C$ depends on the norm of the boundary datum $g_R$).

Along the lines of the proof of [2], Theorem 2.2], making use of the property of the weight functions stated in Eq. (4.2.4) of [26], and of the Sobolev embedding of [26], Lemma 4.2.5], one obtains that, for any ${{\mathbf {x}}}_0\in \Omega$,

$$\begin{aligned} \left| D^{\varvec{\alpha }}u({{\mathbf {x}}}_0)\right|\le & {} C\, k^2 \big (\gamma \,\max \{j+2,k\}\big )^{j+2} \left( \widehat{\Phi }_{j-1,\underline{\beta },{\mathcal {E}}}({{\mathbf {x}}}_0)\right) ^{-1} \nonumber \\\le & {} C\,k^2 \big (\gamma \,\max \{j+2,k\}\big )^{j+2} \left( \widehat{\Phi }_{j,\underline{0},{\mathcal {E}}}({{\mathbf {x}}}_0)\right) ^{-1} \end{aligned}$$

(12)

for all ${\varvec{\alpha }}\in {\mathbb {N}}^2_0$, $|{\varvec{\alpha }}|=j\ge 1$; in the last step we have used the bound $\widehat{\Phi }_{j-1,\underline{\beta },{\mathcal {E}}}({{\mathbf {x}}}_0)\ge \widehat{\Phi }_{j,\underline{0},{\mathcal {E}}}({{\mathbf {x}}}_0)$, which holds true since $0<\beta _\nu <1$; similar $L^\infty$-estimates were derived in [26], Theorem 4.2.23].

Whenever $j\ge k$, $\min \{1,{\mathcal {E}}(j+1)\}=1$, and thus $\widehat{\Phi }_{j,\underline{0},{\mathcal {E}}}({{\mathbf {x}}}_0)=\left( \widehat{\Phi } ({{\mathbf {x}}}_0)\right) ^j$; moreover, $\max \{j+2,k\}=j+2$. By Stirling’s formula, $(j+2)^{j+2}\le 2e^j(j+2)^2j!$ which, for large $j$, gives $(j+2)^{j+2}\le 2(2e)^jj!$. Therefore, we find the following point-wise bounds for all partial derivatives of $u$:

$$\begin{aligned} \left| D^{\varvec{\alpha }}u({{\mathbf {x}}}_0)\right| \le C\,k^2 \left( \frac{2e\,\gamma }{\widehat{\Phi }({{\mathbf {x}}}_0)}\right) ^j \,j!\qquad \forall {\varvec{\alpha }}\in {\mathbb {N}}^2_0,\ |{\varvec{\alpha }}|=j\ge k . \end{aligned}$$

(13)

The analytic continuation to the set in (11) is deduced as in [2], Page 841]. $\square$

Lemma 2.4

With a constant $C>0$ independent of the wave number $k$ (but dependent of the boundary datum $g_R$), the solution $u$ of (1) satisfies

$$\begin{aligned} k\left\| u\right\| _{L^{\infty }(\mathcal {N}(u))} + \left\| \nabla u\right\| _{L^{\infty }(\mathcal {N}(u))} \le C k^{5}\exp (k/4e). \end{aligned}$$

Proof

From (12) and (13), we glean the bounds

$$\begin{aligned}&|D^{{\varvec{\alpha }}}u({{\mathbf {x}}}_{0})| \le Ck^{2}\nonumber \\&\quad \cdot {\left\{ \begin{array}{ll} k^{2} \left( \frac{\gamma k}{\widehat{\Phi }({{\mathbf {x}}}_{0})} \right) ^{j} &{} \; \text {if } |{\varvec{\alpha }}| =: j \le k-2,\\ j! \left( \frac{2e\gamma }{\widehat{\Phi }({{\mathbf {x}}}_{0})} \right) ^{j} &{} \; \text {if } |{\varvec{\alpha }}| =: j > k-2, \end{array}\right. }\quad {\varvec{\alpha }}\in {\mathbb {N}}_{0}^{2},\quad {{\mathbf {x}}}_{0}\in \Omega \setminus \bigcup _{\nu =1}^{n_c} {{\mathbf {c}}}_{\nu }, \end{aligned}$$

(14)

with $C>0$ merely depending on the data $g_{R}$.

For ${{\mathbf {x}}}\in \mathcal {N}(u)$ let ${{\mathbf {x}}}_{0}\in \Omega$ satisfy $\left| {{\mathbf {x}}}-{{\mathbf {x}}}_{0}\right| \le \frac{\widehat{\Phi }({{\mathbf {x}}}_{0})}{4e\gamma }$. The existence of such a point is guaranteed by the definition on $\mathcal {N}(u)$. We have seen that $u({{\mathbf {x}}})$ can be expressed by means of a Taylor series expansion around ${{\mathbf {x}}}_{0}$, which paves the way for the following estimate:

$$\begin{aligned} |u({{\mathbf {x}}})|&\le \left| \sum \limits _{j=0}^{\infty }\frac{1}{j!} D^{j}u({{\mathbf {x}}}_{0})({{\mathbf {x}}}-{{\mathbf {x}}}_{0},\ldots ,{{\mathbf {x}}}-{{\mathbf {x}}}_{0})\right| \le \sum \limits _{j=0}^{\infty } \frac{1}{j!} \max \limits _{|{\varvec{\alpha }}|=j} |D^{{\varvec{\alpha }}}u({{\mathbf {x}}}_{0})| \left| {{\mathbf {x}}}-{{\mathbf {x}}}_{0}\right| ^{j} \\&\overset{(14)}{\le } Ck^{2} \left( \sum \limits _{j=0}^{\lfloor k\rfloor -2} \frac{k^{2}}{j!} \left( \frac{\gamma k\left| {{\mathbf {x}}}-{{\mathbf {x}}}_{0}\right| }{\widehat{\Phi }({{\mathbf {x}}}_{0})} \right) ^{j} + \sum \limits _{j=\lfloor k\rfloor -1}^{\infty } \left( \frac{2e\gamma \left| {{\mathbf {x}}}-{{\mathbf {x}}}_{0}\right| }{\widehat{\Phi }({{\mathbf {x}}}_{0})}\right) ^{j}\right) \\&\le Ck^{2} \left( \sum \limits _{j=0}^{\lfloor k\rfloor -2} \frac{k^{2}}{j!}\left( \frac{k}{4 e} \right) ^{j} + 2^{-(\lfloor k\rfloor -2)} \right) \le Ck^{4}\exp \Big (\frac{k}{4e}\Big ). \end{aligned}$$

The same technique based on a Taylor series shifted by $1$ provides a similar estimate for $\left| \nabla u({{\mathbf {x}}})\right|$. $\square$

3 Trefftz-Discontinuous Galerkin Method

We start from a general mesh ${\mathcal T}_h$ on $\Omega$, whose elements are curvilinear Lipschitz polygons. For any element $K\in {\mathcal T}_h$, we denote by $h_K$ its diameter, and set $h_{\max }:=\max _{K\in {\mathcal T}_h}h_K$. Moreover, we define various sets of interfaces ${{\mathcal F}_h:=\cup _{K\in {\mathcal T}_h}\partial K}$, and ${\mathcal F}_h^I:={\mathcal F}_h\setminus \partial \Omega$.

On the mesh ${\mathcal T}_h$, we introduce the Trefftz space

$$\begin{aligned} T({\mathcal T}_h){:=}\Big \{v\in L^2(\Omega ):\ \exists s>0\ \text {s.t. } v\in H^{\frac{3}{2}+s}({\mathcal T}_h)\ \text {and } \Delta v+k^2v{=}0\ \text {in each } K\in {\mathcal T}_h\Big \}\!, \end{aligned}$$

with $H^{r}({\mathcal T}_h)$ a shorthand notation for elementwise $H^r$-functions on ${\mathcal T}_h$. The solution $u$ of the BVP (1) belongs to $T({\mathcal T}_h)$ and will be approximated in a finite-dimensional Trefftz-DG trial and test space $V_p({\mathcal T}_h)\subset T({\mathcal T}_h)$. At this stage, we need not worry about the details of constructing $V_{p}({\mathcal T}_{h})$; these are postponed to Sect. 6.2.

We fix bounded functions $\alpha ,\beta >0$, $0<\delta \le 1/2$, bounded away from zero and defined on appropriate subsets of ${\mathcal F}_h$. Alluding to the construction of the Trefftz-DG method in [13], Sect. 2], we call them flux parameters. We introduce the following sesquilinear form and antilinear functional defined on $T({\mathcal T}_h)$, cf. [19], Sect. 3.2], [17], Sect. 2], [13], Sect. 2],

$$\begin{aligned} {\mathcal {A}}_h(u,v):=&\int _{{\mathcal F}_h^I}\left( \{\!\{u\}\!\} -\frac{\beta }{ik}[\![\nabla _h u]\!]_N\right) [\![\nabla _h \overline{v}]\!]_N\,\mathrm {d}S\\&\quad -\,\int _{{\mathcal F}_h^I} \Big (\{\!\{\nabla _h u\}\!\}-\alpha ik [\![u]\!]_N\Big ) \cdot [\![\overline{v}]\!]_N\,\mathrm {d}S\\&\quad +\,\int _{\Gamma _R}\left( u-\frac{\delta }{ik\vartheta }\left( \nabla _h u\cdot {{\mathbf {n}}}+ik\vartheta u\right) \right) \left( \overline{\nabla _h v\cdot {{\mathbf {n}}}-ik\vartheta v}\right) \,\mathrm {d}S\\&\quad -\,\int _{\Gamma _D}\Big (\nabla _h u\cdot {{\mathbf {n}}}-\alpha ik u\Big )\overline{v}\,\mathrm {d}S,\\ \ell _{h}(v):=&-\int _{\Gamma _R}\delta \,(ik\vartheta )^{-1}g_R\,\nabla _h \overline{v}\cdot {{\mathbf {n}}}\,\mathrm {d}S +\int _{\Gamma _R}(1-\delta )\,g_R\,\overline{v}\,\mathrm {d}S. \end{aligned}$$

These are the building blocks of the Trefftz-DG variational problem:

$$\begin{aligned} \text {find}\; u_{hp}\in V_p({\mathcal T}_h)\quad \text {such that} \quad {\mathcal {A}}_h(u_{hp},v_{hp}) = \ell _h(v_{hp}) \quad \forall v_{hp}\in V_p({\mathcal T}_h). \end{aligned}$$

(15)

For its analysis, it is convenient to make use of the mesh-dependent DG-norms:

$$\begin{aligned} \left\| v\right\| _{DG}^2 :=&\,k^{-1}\left\| \beta ^{\frac{1}{2}}[\![\nabla _h v]\!]_N\right\| _{0,{\mathcal F}_h^I}^2 +k\left\| \alpha ^{\frac{1}{2}}[\![v]\!]_N\right\| _{0,{\mathcal F}_h^I}^2\\&+\,k^{-1}\left\| \delta ^{\frac{1}{2}}\vartheta ^{-\frac{1}{2}}\nabla _h v\cdot {{\mathbf {n}}}\right\| _{0,\Gamma _R}^2 +k\left\| (1-\delta )^{\frac{1}{2}}\vartheta ^{\frac{1}{2}}v\right\| _{0,\Gamma _R}^2 +k\left\| \alpha ^{\frac{1}{2}}v\right\| _{0,\Gamma _D}^2,\\ \left\| v\right\| _{DG^+}^2:=&\,\left\| v\right\| _{DG}^2+k\left\| \beta ^{-\frac{1}{2}}\{\!\{v\}\!\}\right\| _{0,{\mathcal F}_h^I}^2 +k^{-1}\left\| \alpha ^{-\frac{1}{2}}\{\!\{\nabla _h v\}\!\}\right\| _{0,{\mathcal F}_h^I}^2\\&+\,k\left\| \delta ^{-\frac{1}{2}}\vartheta ^{\frac{1}{2}}v\right\| _{0,\Gamma _R}^2 +k^{-1}\left\| \alpha ^{-\frac{1}{2}}\nabla _h v\cdot {{\mathbf {n}}}\right\| _{0,\Gamma _D}^2. \end{aligned}$$

Here, as in [13, 17, 19], we have used the standard DG notation for averages $\{\!\{\cdot \}\!\}$ and normal jumps $[\![\cdot ]\!]_N$ across interelement boundaries, and $\nabla _h$ designates the elementwise gradient. Since $\alpha$, $\beta$, $\delta$ and $(1-\delta )$ are positive, $\left\| \cdot \right\| _{DG}$ (and thus also $\left\| \cdot \right\| _{DG^+}$) is actually a norm in $T({\mathcal T}_h)$, see [17], Proposition 3.2].

In [19], Propositions 4.1 and 4.3] (see also [17], Sect. 3.1]), we proved the following consistency, continuity and coercivity properties for the variational problem (15): for $u$ solution of the BVP (1) and for all $v,w\in T({\mathcal T}_h)$

$$\begin{aligned} {\mathcal {A}}_h(u,v) = \ell _h(v),\quad \left| {\mathcal {A}}_{h}(v,w)\right| \le 2\left\| v\right\| _{DG^+}\left\| w\right\| _{DG},\quad {{\mathrm{Im}}}[{\mathcal {A}}_{h}(v,v)]=\left\| v\right\| _{DG}^2. \end{aligned}$$

This ensures that (15) is well posed, stable and that the Trefftz-DG method enjoys quasi-optimality in the DG-norm, i.e.

$$\begin{aligned} \left\| u-u_{hp}\right\| _{DG} \le 3\inf _{v_{hp}\in V_p({\mathcal T}_h)}\left\| u-v_{hp}\right\| _{DG^+}, \end{aligned}$$

(16)

where $u_{hp}$ is the solution of the discrete variational problem (15). The Trefftz-DG is therefore unconditionally stable, i.e. the quasi-optimality bound (16) holds with the same constant for any wavenumber $k>0$, any mesh ${\mathcal T}_h$, any discrete Trefftz space $V_p({\mathcal T}_h)$ and any admissible choice of the flux parameters; on the other hand, the DG-norms used to measure the error in (16) depend on $k,{\mathcal T}_h,\alpha ,\beta$ and $\delta$ (but not on the specific discrete space $V_p({\mathcal T}_h)$).

Remark 3.1

In the case of homogeneous Neumann boundary conditions along the interior boundary now denoted by $\Gamma _N$ (scattering by a sound-hard material), the bilinear form ${\mathcal {A}}_h(\cdot ,\cdot )$ in the formulation (15) contains the term $\int _{\Gamma _N}\big (u-\beta (ik)^{-1}\nabla _h u\cdot {{\mathbf {n}}}\big )\nabla _h\overline{v}\cdot {{\mathbf {n}}}\,\mathrm {d}S$ instead of ${-\int _{\Gamma _D}\big (\nabla _h u\cdot {{\mathbf {n}}}-\alpha ik u\big )\overline{v}\,\mathrm {d}S}$. Wavenumber-explicit stability and regularity results for solutions in this case, analogue to the one for the Dirichlet case discussed in Sect. 2, are not available at present.

4 $L^{2}$-Estimates

Our principal goal is to study the convergence of the discretisation error of the Trefftz-DG method not only in the mesh-dependent DG-norm $\left\| \cdot \right\| _{DG}$, but also in the $L^{2}(\Omega )$-norm. This is made possible by a key duality technique originally introduced in [35], Theorem 3.1] and improved in [17], Sect. 3.2] and [19], Sect. 4.2]. In Lemma 4.5, we further modify this duality argument to allow for different flux parameters.

4.1 Assumptions on the Meshes

We study the convergence of Trefftz-DG methods for an infinite family of meshes $\mathfrak {T} := \left\{ {\mathcal T}_h\right\}$ whose members enjoy certain properties uniformly:

(M1) :

star-shapedness: there exist $0<\rho _0<\rho \le 1/2$ such that, for all the meshes ${\mathcal T}_{h}\in \mathfrak {T}$ and for all $K\in {\mathcal T}_h$, there exists ${{\mathbf {x}}}_K\in K$ such that $B_{\rho h_K}({{\mathbf {x}}}_K)\subset K$, and $K$ is star-shaped with respect to $B_{\rho _0 h_K}({{\mathbf {x}}}_K)$;

(M2) :

local quasi-uniformity: there exists a constant $\tau \ge 1$ such that, for all the meshes ${\mathcal T}_{h}\in \mathfrak {T}$,

$$\begin{aligned} \tau ^{-1}\le \frac{h_{K_1}}{h_{K_2}}\le \tau \qquad \forall K_1,K_2\in {\mathcal T}_h \text { s.t. } |\partial K_1\cap \partial K_2|\ne 0; \end{aligned}$$

(M3) :

boundedness of the skeleton measure: there exists a constant $C_{{\mathcal F}}>0$ such that, for all the meshes ${\mathcal T}_{h}\in \mathfrak {T}$,

$$\begin{aligned} |{\mathcal F}_h^I|\le C_{{\mathcal F}}. \end{aligned}$$

Here and in the following, we adopt the notation $\left| \cdot \right|$ for the volume (area or length) of one- or two-dimensional sets. Assumptions (M1)–(M3) are instrumental for achieving abstract error estimates in the $L^{2}(\Omega )$-norm in Sect. 4.4. In Sect. 6.1, they will be supplemented with more specific requirements for $hp$-approximation.

An important tool is the similarity transformation ${{\mathbf {x}}}\mapsto \hat{{{\mathbf {x}}}} := h_{K}^{-1}({{\mathbf {x}}}-{{\mathbf {x}}}_{K})$, which takes an element $K\in {\mathcal T}_{h}$ to a domain $\hat{K}$ with ${\text {diam}}(\hat{K})=1$, which contains $B_{\rho }$ and is star-shaped with respect to the ball $B_{\rho _{0}}$.

4.2 Flux Parameters

We still have the freedom to fix the so-called flux parameters $\alpha ,\beta ,\delta$ entering ${\mathcal {A}}_{h}$ and $\ell _{h}$. Linking them to the local mesh width in a judicious fashion was essential for coping with locally refined meshes in [19]. Hardly surprising, the right choice of the flux parameters is also key to a successful analysis of the $hp$-version of the Trefftz-DG method. It differs slightly from what was used in [19], Formula (21)].

We fix the function $\alpha$ on any face $f\subset {\mathcal F}_h^I\cup \Gamma _D$ as follows:

$$\begin{aligned} \alpha |_f:=\mathtt{a}\,\frac{h_{{\max }}}{h_f}, \end{aligned}$$

(17)

where $\mathtt{a}$ is a positive universal constant, in particular independent of the local mesh sizes, the local Trefftz spaces, and the wavenumber $k$. The symbol $h_{f}$ stands for the local mesh width at the interface $f$ defined as

$$\begin{aligned} h_{f}:={\left\{ \begin{array}{ll} \min \{h_{K_1},h_{K_2}\} &{} \text {if } f=\partial K_1\cap \partial K_2,\\ h_K &{} \text {if } f=\partial K\cap \partial \Omega . \end{array}\right. } \end{aligned}$$

Notice that this definition works also in the case of hanging nodes (compare with Assumption (M2)). Moreover, we choose

$$\begin{aligned} \beta ,\;\delta \quad \text {as fixed positive} \ { universal \ constants}, \end{aligned}$$

(18)

of course, with the additional constraint $\delta \le 1/2$.

Remark 4.1

The choice of $\beta$ and $\delta$ independent of the local mesh sizes, as opposed to $\beta |_f,\delta |_f\simeq \frac{h_{{\max }}}{h_{f}}$ as in [19], ensures that the coefficients in front of the gradient terms in the $DG$-norm do not blow up in regions where the mesh is refined. This permits us to accomplish convergence estimates on strongly locally refined meshes in Sect. 6. To that end, in Sect. 4.4, we modify the duality argument of [19].

Remark 4.2

The orders of $h$- and $p$-convergence of the Trefftz-discontinuous Galerkin method posed on quasi-uniform meshes are identical to those presented in [17, 19], since for these meshes all flux parameters $\alpha$, $\beta$ and $\delta$ are constant. To improve the orders of convergence in $h$, the parameters of [13] may be used.

4.3 Trace Inequalities

As technical tools, we use the following trace inequalities:

$$\begin{aligned} \left\| v\right\| _{0,\partial K}^2&\le C_1\Big (h_K^{-1}\left\| v\right\| _{0,K}^2+h_K\left| v\right| _{1,K}^2\Big ) \quad&\forall v\in H^1(K), \end{aligned}$$

(19)

$$\begin{aligned} \left\| \nabla v\right\| _{0,\partial K}^2&\le C_2\Big (h_K^{-1}\left\| \nabla v\right\| _{0,K}^2+h_K^{2s}\left| \nabla v\right| _{\frac{1}{2}+s,K}^2\Big ) \quad&\forall v\in H^{\frac{3}{2}+s}(K), \;\, s\in (0,1/2), \end{aligned}$$

(20)

where $C_1$ depends only on $\rho _0$, and $C_2$ on $\rho _0$, $\rho$ and $s$. Taking $v=1$ in (19), we can also see that

$$\begin{aligned} \left| \partial K\right| \le C_1\,h_K, \end{aligned}$$

(21)

with the same $C_1$ as above, depending only on $\rho _0$.

Remark 4.3

The dependences of the constants show that the parameters $\rho$, $\rho _0$ and $h_K$ capture all the geometrical information that is relevant for the trace inequalities, since both the “roughness” of $\partial K$ (i.e. its Lipschitz constant in some parametrisation) and the “fatness” of $K$ (i.e. the maximal distance of the interior points from the boundary and the relation between its measure and that of its boundary) are controlled by their values.

The bound (19) is standard (see, e.g. [5], Theorem (1.6.6)]), while (20), for simplicial elements, can be proved using [27], Theorem A.2]. Under our Assumption (M1) on the star-shapedness of the mesh element $K$, the trace inequalities (19) and  (20), with explicit dependence of the constants on $\rho$, $\rho _0$ and $s$, readily follow from the following lemma by scaling arguments.

Lemma 4.4

Let $\hat{K}\subset \mathbb {R}^2$ be such that ${{\mathrm{diam}}}(\hat{K})=1$ and let there exist $0<\rho _0<\rho \le 1/2$ such that $B_{\rho }\subset \hat{K}$, and $\hat{K}$ is star-shaped with respect to $B_{\rho _0}$. Then,

$$\begin{aligned}&\left\| v\right\| _{0,\partial \hat{K}}^2\le \frac{1+\sqrt{2}}{\rho _0} \big (\left\| v\right\| _{0,\hat{K}}^2+\left| v\right| _{1,\hat{K}}^2\big ) \qquad \qquad \qquad \qquad \,\, \forall v\in H^1(\hat{K}), \end{aligned}$$

(22)

$$\begin{aligned} \left\| w\right\| _{0,\partial \hat{K}}^2&\le C_{B_1}\frac{1}{\rho ^2}\left( \frac{3}{\rho _0\rho ^2}\right) ^{4+2s} \big (\left\| w\right\| _{0,K}^2+\left| w\right| _{\frac{1}{2}+s,\hat{K}}^2\big )&\forall w\in H^{\frac{1}{2}+s}(\hat{K}), \quad s\in (0,1/2), \end{aligned}$$

(23)

where $C_{B_1}$ depends on $s$ but not on $\hat{K}$.

Proof

We start with (22). Denoting by ${{\mathbf {n}}}_K$ the outward normal unit vector to $\partial \hat{K}$, since $\hat{K}$ is star-shaped with respect to $B_{\rho _0}$, we have

$$\begin{aligned} {{\mathbf {n}}}_K({{\mathbf {x}}})\cdot {{\mathbf {x}}}\ge \rho _0 \qquad \text {a.e. on } \partial \hat{K}, \end{aligned}$$

(24)

where the inequality is meant to hold for every point ${{\mathbf {x}}}$ at which ${{\mathbf {n}}}_K({{\mathbf {x}}})$ is defined (see [18], Lemma 3.1]). Thus,

$$\begin{aligned} \left\| v\right\| _{0,\partial \hat{K}}^2= & {} \int _{\partial \hat{K}}\left| v\right| ^2\,\mathrm {d}S\\&\overset{(24)}{\le }\frac{1}{\rho _0}\int _{\partial \hat{K}}{{\mathbf {n}}}_K\cdot {{\mathbf {x}}}\left| v\right| ^2\,\mathrm {d}S\\= & {} \frac{1}{\rho _0}\int _{\hat{K}}{\text {div}}({{\mathbf {x}}}\left| v\right| ^2)\,\mathrm {d}{{\mathbf {x}}}= \frac{1}{\rho _0}\int _{\hat{K}} (2\left| v\right| ^2+{{\mathbf {x}}}\cdot \nabla \left| v\right| ^2)\,\mathrm {d}{{\mathbf {x}}}\\= & {} \frac{1}{\rho _0}\int _{\hat{K}}(2\left| v\right| ^2+2{{\mathbf {x}}}\cdot {{\mathrm{Re}}}\{v\,\nabla \overline{v}\})\,\mathrm {d}{{\mathbf {x}}}\\&\overset{\begin{array}{c} {{\mathrm{diam}}}(\hat{K})=1\\ \Rightarrow \,\left| {{\mathbf {x}}}\right| \le 1 \end{array}}{\le }\frac{2}{\rho _0} \left( \left\| v\right\| _{0,\hat{K}}^2+\left\| v\right\| _{0,\hat{K}}\left\| \nabla v\right\| _{0,\hat{K}}\right) \\\le & {} \frac{2}{\rho _0} \left( \left\| v\right\| _{0,\hat{K}}^2+\frac{1}{2(1+\sqrt{2})}\left\| v\right\| _{0,\hat{K}}^2 +\frac{(1+\sqrt{2})}{2}\left\| \nabla v\right\| _{0,\hat{K}}^2\right) \\= & {} \frac{1+\sqrt{2}}{\rho _0}\left( \left\| v\right\| _{0,\hat{K}}^2+\left\| \nabla v\right\| _{0,\hat{K}}^2\right) , \end{aligned}$$

which gives (22).

For the bound (23), we recall Assumption (M1) and, without loss of generality, place the centre of $K$ at the origin, that is, ${{\mathbf {x}}}_{K}={\mathbf {0}}$. We identify $\mathbb {R}^2$ and $\mathbb {C}$ and make use of the polar parametrisation $\Psi :\mathbb {C}\rightarrow \mathbb {C}$ such that

$$\begin{aligned} \Psi (B_1)=\hat{K},\qquad \Psi (re^{i\theta })=\psi (\theta )re^{i\theta },\qquad \psi :[-\pi ,\pi )\rightarrow [\rho ,1-\rho ]. \end{aligned}$$

The function $\psi$ is Lipschitz continuous with constant $L_\psi$ satisfying

$$\begin{aligned} L_\psi =\sup _{\theta \in [-\pi ,\pi ]}\psi ^\prime (\theta )\le \frac{(1-\rho )^2}{\rho _0} \end{aligned}$$

(25)

(see [20], Lemma 4.1]), and the function $\Psi ^{-1}:\mathbb {C}\rightarrow \mathbb {C}$ is Lipschitz continuous as well, with constant $L_{\Psi ^{-1}}$ satisfying

$$\begin{aligned} L_{\Psi ^{-1}} =\sup _{w,v\in \mathbb {C},w\ne v}\frac{\left| w-v\right| }{\left| \Psi (w)-\Psi (v)\right| } \le \frac{2(2\rho +L_\psi )}{\rho ^2} \end{aligned}$$

(26)

(see [20], Lemma 4.2]).

We have

$$\begin{aligned} \left\| w\right\| _{0,\partial \hat{K}}^2&=\int _{\partial \Psi (B_1)}\left| w\right| ^2\,\mathrm {d}S=\int _{-\pi }^{\pi }\left| (w\circ \Psi )(e^{i\theta })\right| ^2\left| \psi ^\prime (\theta )\right| \,\mathrm {d}\theta \\&\le L_\psi \int _{-\pi }^{\pi }\left| (w\circ \Psi )(e^{i\theta })\right| ^2 \,\mathrm {d}\theta \overset{(25)}{\le }\frac{(1-\rho )^2}{\rho _0}\int _{-\pi }^{\pi }\left| (w\circ \Psi )(e^{i\theta })\right| ^2\,\mathrm {d}\theta \\&= \frac{(1-\rho )^2}{\rho _0} \left\| w\circ \Psi \right\| _{0,\partial B_1}^2 {\le } \frac{(1-\rho )^2}{\rho _0} C_{B_1} \left( \left\| w\circ \Psi \right\| _{0,B_1}^2+\left| w\circ \Psi \right| _{\frac{1}{2}+ s ,B_1}^2\right) , \end{aligned}$$

where the last inequality can be proved using [27], Theorem A.2]; clearly, the constant $C_{B_1}$, which corresponds to that appearing in the analogous of the trace inequality (23) for the unit ball $B_1$, depends on $s$ and not on $\hat{K}$.

By definition of the $(\frac{1}{2}+ s )$-seminorm by the Sobolev–Slobodeckij integral, the Lipschitz property of $\Psi ^{-1}$, and by changing variables within integrals, we obtain

$$\begin{aligned} \left| w\circ \Psi \right| _{\frac{1}{2}+ s ,B_1}^2&= \int _{B_1}\int _{B_1}\frac{\left| (w\circ \Psi )({{\mathbf {x}}}_{B})-(w\circ \Psi )({{\mathbf {y}}}_{B})\right| ^2}{\left| {{\mathbf {x}}}_{B}-{{\mathbf {y}}}_{B}\right| ^{3+2 s }}\,\mathrm {d}{{\mathbf {x}}}_B \,\mathrm {d}{{{\mathbf {y}}}}_B\\&\le \int _{B_1}\int _{B_1}L_{\Psi ^{-1}}^{3+2 s } \frac{\left| (w\circ \Psi )({{\mathbf {x}}}_{B})-(w\circ \Psi )({{\mathbf {y}}}_{B})\right| ^2}{\left| \Psi ^{-1}({{\mathbf {x}}}_{B})-\Psi ^{-1}({{\mathbf {y}}}_{B})\right| ^{3+2 s }} \,\mathrm {d}{{\mathbf {x}}}_B \,\mathrm {d}{{{\mathbf {y}}}}_B\\&\le \int _{\hat{K}}\int _{\hat{K}} L_{\Psi ^{-1}}^{3+2 s }\frac{\left| w({{\mathbf {x}}})-w({{\mathbf {y}}})\right| ^2}{\left| {{\mathbf {x}}}-{{\mathbf {y}}}\right| ^{3+2 s }} \left| \det D\Psi ^{-1}{({{\mathbf {x}}})}\right| {\left| \det D\Psi ^{-1}({{\mathbf {y}}})\right| }\,\mathrm {d}{{\mathbf {x}}}\,\mathrm {d}{{\mathbf {y}}}, \end{aligned}$$

and

$$\begin{aligned} \left\| w\circ \Psi \right\| _{0,B_1}^2 =\int _{B_1}\left| w\circ \Psi {({{\mathbf {x}}}_B)}\right| ^2\,\mathrm {d}{{\mathbf {x}}}_B =\int _{\hat{K}}\left| w\right| ^2 \left| \det {D\Psi ^{-1}}{({{\mathbf {x}}})}\right| \,\mathrm {d}{{\mathbf {x}}}. \end{aligned}$$

From the expression of the Jacobian $D\Psi ^{-1}$ in Cartesian coordinates given in the proof of [20], Lemma 4.2], we compute

$$\begin{aligned} \left| \det {D\Psi ^{-1}}\right| =\frac{1}{\psi (\theta )^2}\le \frac{1}{\rho ^2}. \end{aligned}$$

Therefore,

$$\begin{aligned} \left\| w\right\| _{0,\partial \hat{K}}^2\le & {} {C_{B_1}}\frac{(1-\rho )^2}{\rho _0\rho ^2}\left( \left\| w\right\| _{0,\hat{K}}^2 +\frac{L_{\Psi ^{-1}}^{3+2 s }}{{\rho ^2}}\left| w\right| _{\frac{1}{2}+ s ,\hat{K}}^2\right) \\&\overset{(26), (25)}{\le }{C_{B_1}}\frac{(1-\rho )^2}{\rho _0\rho ^2} \left( \left\| w\right\| _{0,\hat{K}}^2+{\frac{1}{\rho ^2}} \left( \frac{3}{\rho _0\rho ^2}\right) ^{3+2 s }\left| w\right| _{\frac{1}{2}+ s ,\hat{K}}^2\right) , \end{aligned}$$

from which, we get (23). $\square$

4.4 Duality Argument

By using a similar argument as in [6, 17, 19, 35], we bound the $L^2$-norm of any Trefftz function by its $DG$-norm, with explicit dependence of the bounding constant on the wavenumber. The first part of the proof of the following lemma is identical to that of [19], Lemma 4.4]. We report the whole proof for completeness.

Lemma 4.5

For any $\varepsilon >0$, there exists a constant $C>0$ depending only on the shape of $\Omega$, $\vartheta$, $\rho _0$, $\rho$, $\mathtt{a}$, $\beta$, $\delta$ and $\varepsilon$ (in particular independent of $V_{p}({\mathcal T}_{h})$, ${\mathcal T}_{h}$ and $k$) such that, for any $w\in T({\mathcal T}_h)$,

$$\begin{aligned} \left\| w\right\| _{0,\Omega }\le C\, \left[ \frac{1}{kh_{\max }}+ k^{1+2\varepsilon }(C_{\mathcal F}+\left| \Gamma _R\right| )\right] ^{\frac{1}{2}} \left\| w\right\| _{DG}. \end{aligned}$$

Proof

Let $\phi$ be in $L^2(\Omega )$. Let $v$ be the solution to the (adjoint) problem (2)–(4) with $f=\phi$, $g_R=0$ and “$-$” in the impedance condition on $\Gamma _R$. From Proposition 2.1, we know that $v\in H^{1+t}(\Omega )$ for all $0\le t<1/2+s_\Omega$ (with $s_\Omega$ defined in Proposition 2.1) and that

$$\begin{aligned} |v|_{1,\Omega }+k\left\| v\right\| _{0,\Omega }\le C\left\| \phi \right\| _{0,\Omega },\qquad \left| \nabla v\right| _{t,\Omega } \le C \,(1+ k^t)\left\| \phi \right\| _{0,\Omega }, \end{aligned}$$

(27)

with $C>0$ depending only on $s$, $\gamma _R$ and $\vartheta$, but independent of $k$, $\phi$ and $v$. In particular, $v\in H^{\frac{3}{2}+s}(\Omega )$ for all $0<s<s_\Omega$.

Multiplying by $w\in T({\mathcal T}_h)$, integrating by parts twice the Eq. (2) element by element (using $\Delta w+k^2 w=0$ in each $K\in {\mathcal T}_h$), and taking into account that $\nabla v\cdot {{\mathbf {n}}}=ik\vartheta v$ on $\Gamma _R$ and $v=0$ on $\Gamma _D$, we obtain

$$\begin{aligned} |(w,\phi )_{0,\Omega }|= & {} \left| \sum _{K\in {\mathcal T}_h}\int _{\partial K} \left( \nabla w\cdot {{\mathbf {n}}}\,\overline{v}-w\,\overline{\nabla v\cdot {{\mathbf {n}}}}\right) \,\mathrm {d}S\right| \\= & {} \left| \int _{{\mathcal F}_h^I}\left( [\![\nabla _h w]\!]_N\overline{v}-[\![w]\!]_N\cdot \overline{\nabla v}\right) \,\mathrm {d}S \right. \left. +\int _{\Gamma _R}\left( \nabla _h w\cdot {{\mathbf {n}}}+ik {\vartheta } w\right) \,\overline{v}\,\mathrm {d}S \right. \\&\left. -\,\int _{\Gamma _D}w\,\overline{\nabla v}\cdot {{\mathbf {n}}}\,\mathrm {d}S\right| , \end{aligned}$$

from which, by the Cauchy–Schwarz inequality,

$$\begin{aligned} |(w,\phi )_{0,\Omega }| \le&\sum _{f\subset {\mathcal F}_h^I}\left( k^{-\frac{1}{2}}\left\| \beta ^{\frac{1}{2}}[\![\nabla _h w]\!]_N\right\| _{0,f} k^{\frac{1}{2}}\left\| \beta ^{-\frac{1}{2}}v\right\| _{0,f}\right. \\&\quad \left. +\,k^{\frac{1}{2}}\left\| \alpha ^{\frac{1}{2}}[\![w]\!]_N\right\| _{0,f} k^{-\frac{1}{2}}\left\| \alpha ^{-\frac{1}{2}}\nabla v\cdot {{\mathbf {n}}}\right\| _{0,f} \right) \\&\quad + \sum _{f\subset \Gamma _R}\left( k^{-\frac{1}{2}}\left\| \delta ^{\frac{1}{2}}\vartheta ^{-\frac{1}{2}}\nabla {w}\cdot {{\mathbf {n}}}\right\| _{0,f} k^{\frac{1}{2}}\left\| \delta ^{-\frac{1}{2}}\vartheta ^{\frac{1}{2}} v\right\| _{0,f}\right. \\&\quad \left. +\,k^{\frac{1}{2}}\left\| \delta ^{\frac{1}{2}}\vartheta ^{\frac{1}{2}}{w}\right\| _{0,f} k^{\frac{1}{2}}\left\| \delta ^{-\frac{1}{2}}\vartheta ^{\frac{1}{2}} v\right\| _{0,f} \right) \\&\quad + \sum _{f\subset \Gamma _D} k^{\frac{1}{2}}\left\| \alpha ^{\frac{1}{2}}w\right\| _{0,f} k^{-\frac{1}{2}}\left\| \alpha ^{-\frac{1}{2}}\nabla v\cdot {{\mathbf {n}}}\right\| _{0,f}\\ \le&\, \left\| w\right\| _{DG} {\mathcal {G}}(v)^{\frac{1}{2}}, \end{aligned}$$

where we have set

$$\begin{aligned} {\mathcal {G}}(v):&=\sum _{f\subset {\mathcal F}_h^I}\left( k\left\| \beta ^{-\frac{1}{2}}v\right\| _{0,f}^2+k^{-1}\left\| \alpha ^{-\frac{1}{2}}\nabla v\cdot {{\mathbf {n}}}\right\| _{0,f}^2\right) \\&\quad + \sum _{f\subset \Gamma _R}2k\left\| \delta ^{-\frac{1}{2}}\vartheta ^{\frac{1}{2}} v\right\| _{0,f}^2 + \sum _{f\subset \Gamma _D}k^{-1}\left\| \alpha ^{-\frac{1}{2}}\nabla v\cdot {{\mathbf {n}}}\right\| _{0,f}^2. \end{aligned}$$

We need to bound ${\mathcal {G}}(v)$ in terms of $\left\| \phi \right\| ^2_{0,\Omega }$. We exploit the fact that $v\in L^{\infty }(\Omega )$, together with the Assumption (M3) on the mesh family, in order to bound the terms containing $\beta$ and $\delta$. Since $\nabla v$ does not necessarily belong to $L^{\infty }(\Omega )$, we cannot use the same argument for the terms containing $\alpha$. We report, for completeness, the estimate of the terms containing $\alpha$ from [19].

Using the trace inequality (20) and taking into account the local quasi-uniformity Assumption (M2), we obtain

$$\begin{aligned} \sum _{f\subset {\mathcal F}_h^I} k^{-1}&\left\| \alpha ^{-\frac{1}{2}}\nabla v\cdot {{\mathbf {n}}}\right\| _{0,f}^2 +\sum _{f\subset \Gamma _D}k^{-1}\left\| \alpha ^{-\frac{1}{2}}\nabla v\cdot {{\mathbf {n}}}\right\| _{0,f}^2\\&\overset{(20)}{\le }C\, \sum _{K\in {\mathcal T}_h} \left\| \alpha ^{-\frac{1}{2}}\right\| _{L^\infty \left( \partial K\cap ({\mathcal F}_h^I\cup \Gamma _D)\right) }^2 \left[ \frac{1}{kh_{K}}\left\| \nabla v\right\| _{0,K}^{2} + \frac{h_{K}^{2{s}}}{k}\left| \nabla v\right| _{\frac{1}{2}+{s},K}^{2} \right] , \end{aligned}$$

with $C>0$ depending only on $\rho _0$, $\rho$ and $s$. The Assumption (17) on $\alpha$ implies

$$\begin{aligned} \left\| \alpha ^{-\frac{1}{2}}\right\| _{L^\infty \left( \partial K\cap ({\mathcal F}_h^I\cup \Gamma _D)\right) }^2 \le \frac{h_K}{\mathtt{a}\,h_{{\max }}}, \end{aligned}$$

which leads to the estimate

$$\begin{aligned} \sum _{f\subset {\mathcal F}_h^I}k^{-1}&\left\| \alpha ^{-\frac{1}{2}}\nabla v\cdot {{\mathbf {n}}}\right\| _{0,f}^2 +\sum _{f\subset \Gamma _D}k^{-1}\left\| \alpha ^{-\frac{1}{2}}\nabla v\cdot {{\mathbf {n}}}\right\| _{0,f}^2\\&\overset{(20)}{\le }C\, \sum _{K\in {\mathcal T}_h} \left[ \frac{1}{kh_{{\max }}}\left\| \nabla v\right\| _{0,K}^{2} + \frac{h_{K}^{2{s}+1}}{kh_{{\max }}}\left| \nabla v\right| _{\frac{1}{2}+{s},K}^{2}\right] , \end{aligned}$$

where, now, $C$ also depends on $\mathtt{a}$. By definition, $h_{K}\le h_{{\max }}$, and therefore (27) (taken with $t=1/2+s$) gives

$$\begin{aligned}&\sum _{f\subset {\mathcal F}_h^I}k^{-1} \left\| \alpha ^{-\frac{1}{2}}\nabla v\cdot {{\mathbf {n}}}\right\| _{0,f}^2 +\sum _{f\subset \Gamma _D}k^{-1}\left\| \alpha ^{-\frac{1}{2}}\nabla v\cdot {{\mathbf {n}}}\right\| _{0,f}^2\nonumber \\&\quad \le C\, \big (k^{-1}h_{{\max }}^{-1}+{}(kh_{\max })^{2{s}}\big )\left\| \phi \right\| _{0,\Omega }^2. \end{aligned}$$

(28)

We proceed now with the estimate of the terms in ${\mathcal {G}}(v)$ containing $\beta$ and $\delta$. Let us start with the term containing $\beta$. From the Sobolev embedding $H^{1+\varepsilon }(\Omega )\subset C^0(\overline{\Omega })$, for any $\varepsilon >0$ (see e.g. [24], Theorem 3.26]), we have $v\in L^\infty (\Omega )$ and

$$\begin{aligned} \sum _{f\subset {\mathcal F}_h^I} k\left\| \beta ^{-\frac{1}{2}}v\right\| _{0,f}^2&\le k\, \left| {\mathcal F}_h^I\right| \left\| \beta ^{-\frac{1}{2}}v\right\| _{L^\infty ({\mathcal F}_h^I)}^2 \le \beta ^{-1}k\, \left| {\mathcal F}_h^I\right| \left\| v\right\| _{L^\infty (\Omega )}^2, \end{aligned}$$

Provided that $\varepsilon <1/2+s_\Omega$, $v\in H^{1+\varepsilon }(\Omega )$, and there exists $C>0$ depending only on the shape of $\Omega$ and $\varepsilon$ such that

$$\begin{aligned} \left\| v\right\| _{L^\infty (\Omega )}^2 \le C \Big (\left\| v\right\| _{0,\Omega }^2+\left\| \nabla v\right\| _{0,\Omega }^2 +\left| \nabla v\right| _{\varepsilon ,\Omega }^2\Big ). \end{aligned}$$

By using (27) with $t=\varepsilon$, we obtain

$$\begin{aligned} \left\| v\right\| _{L^\infty (\Omega )}^2\le C \left( \frac{1}{k^2}+1+k^{2\varepsilon } \right) \left\| \phi \right\| _{0,\Omega }^2, \end{aligned}$$

and thus

$$\begin{aligned} \sum _{f\subset {\mathcal F}_h^I} k\left\| \beta ^{-\frac{1}{2}}v\right\| _{0,f}^2\le C\, \left| {\mathcal F}_h^I\right| \left( k^{-1}+ k^{1+2\varepsilon }\right) \left\| \phi \right\| _{0,\Omega }^2, \end{aligned}$$

(29)

with $C$ only depending on the shape of $\Omega$, $\vartheta$, $\varepsilon$ and $\beta$.

We bound the term containing $\delta$ similarly:

$$\begin{aligned} \sum _{f\subset \Gamma _R}2k\left\| \delta ^{-\frac{1}{2}}\vartheta ^{\frac{1}{2}} v\right\| _{0,f}^2&\le 2k\,\left| \Gamma _R\right| \left\| \delta ^{-\frac{1}{2}}\vartheta ^{\frac{1}{2}} v\right\| _{L^\infty (\Gamma _R)}^2\nonumber \\&\le 2k\,\delta ^{-1}\left\| \vartheta \right\| _{L^\infty (\Gamma _R)} \left| \Gamma _R\right| \left\| v\right\| _{L^\infty (\Omega )}^2,\nonumber \\&\le C\,\left| \Gamma _R\right| \left( k^{-1}+ k^{1+2{\varepsilon }}\right) \left\| \phi \right\| _{0,\Omega }^2, \end{aligned}$$

(30)

with $C$ only depending on the shape of $\Omega$, $\vartheta$, $\varepsilon$ and $\delta$.

Thus, collecting the bounds (28)–(30) on the terms containing $\alpha$, $\beta$ and $\delta$ in the definition of ${\mathcal {G}}(v)$, for all $\phi \in L^2(\Omega )$, we have

$$\begin{aligned} {\mathcal {G}}(v)\le C\,\left( k^{-1}h_{\max }^{-1} + k^{2s}+ \left| {\mathcal F}_h^I\cup \Gamma _R\right| (k^{-1}+ k^{1+2\varepsilon }) \right) \left\| \phi \right\| _{0,\Omega }^2, \end{aligned}$$

and thus, due to Assumption (M3) and $2s<1$,

$$\begin{aligned} \frac{\left| (w,\phi )_{0,\Omega }\right| }{\left\| \phi \right\| _{0,\Omega }}\le C\, \left[ \frac{1}{kh_{\max }}+ k^{1+2\varepsilon }(C_{\mathcal F}+\left| \Gamma _R\right| )\right] ^{\frac{1}{2}}\left\| w\right\| _{DG}. \end{aligned}$$

For larger values of $\varepsilon$, the same bound holds. $\square$

We note that in the assertion of Lemma 4.5, we can take an arbitrarily small $\varepsilon >0$ to reduce the dependence on $k$, but the constant $C$ may blow up in the limit $\varepsilon \rightarrow 0$ as it contains the continuity constant of the embedding of $H^{1+\varepsilon }(\Omega )$ in $L^\infty (\Omega )$.

Since $u-u_{hp}\in T({\mathcal T}_h)$, from Lemma 4.5 and the quasi-optimality (16), we immediately deduce the following result.

Theorem 4.6

Assume the mesh properties (M1)–(M3) and that the solution $u$ of (1) belongs to $T({\mathcal T}_h)$, and let $u_{hp}$ be the solution of (15). Then, for any $\varepsilon >0$, there exists a constant $C>0$ depending only on the shape of $\Omega$, $\vartheta$, $\rho _0$, $\rho$, $\mathtt{a}$, $\beta$, $\delta$ and $\varepsilon$ (in particular independent of $V_{p}({\mathcal T}_{h})$, ${\mathcal T}_{h}$ and $k$) such that

$$\begin{aligned} \left\| u-u_{hp}\right\| _{0,\Omega } \le C\, \left[ \frac{1}{kh_{\max }}+ k^{1+2{\varepsilon }}(C_{\mathcal F}+\left| \Gamma _R\right| )\right] ^{\frac{1}{2}} \inf _{v_{hp}\in V_p({\mathcal T}_h)} \left\| u-v_{hp}\right\| _{DG^+}. \end{aligned}$$

5 Approximation Properties of Plane Wave Spaces

In this section, we consider a Helmholtz solution $u$ defined in the neighbourhood

$$\begin{aligned} K_\eta :=\big \{{{\mathbf {x}}}\in \mathbb {R}^2,\, {{\mathrm{dist}}}({{\mathbf {x}}},K)<\eta h_K\big \},\qquad 0<\eta \le 1/2, \end{aligned}$$

of an (open) element $K$ satisfying the star-shapedness Assumption (M1); for simplicity, we take $K$ to be centred at the origin, i.e. $B_{\rho h_K}\subset K$ and ${{\mathbf {n}}}({{\mathbf {x}}})\cdot {{\mathbf {x}}}\ge \rho _0 h_K$ a.e. on $\partial K$. We note that $K_\eta$ contains $B_{(\rho +\eta )h_K}$ and is star-shaped with respect to $B_{(\rho _0+\eta )h_K}$. Following the theory developed in [20, 30, 31] we prove approximation bounds for finite-dimensional spaces made of circular and plane wave functions.

The main ingredients are three: (i) the explicit approximation bounds for harmonic functions and harmonic polynomials proved in [20] (improving on [25]) and reported in Proposition 5.1; (ii) the Vekua operators, which permit to transfer these approximation properties to Helmholtz solutions and circular waves (see a detailed discussion [32] and the continuity bounds in Lemma 5.2 below); (iii) the approximate inversion of the Jacobi–Anger expansion, which allows to prove bounds for plane waves (see (39) below, which was proved in [31], Lemma 4.3]). The interplay of these ingredients is outlined in Fig. 1.

Fig. 1

The idea behind the approximation estimates of Sect. 5: plane waves approximate circular waves (Fourier–Bessel functions), which are Vekua transforms of harmonic polynomials, which approximate harmonic functions, which in turn are inverse Vekua transforms of Helmholtz solutions. The $\rightarrow$ arrow denotes the Vekua operators, which are bijective mappings, and the $\twoheadrightarrow$ arrow can be read as “is approximated by”; the curved arrows are consequences of the straight ones

We consider only $W^{j,\infty }$-type norms (as opposed to $H^j$-type) in our bounds; moreover, since $u$ is analytic in $K_\eta$, its possible singularities lie at least at distance $\eta$ from $K$: these two facts make the proofs easier than those in [31] (even though here we obtain exponential convergence as opposed to algebraic). On the other hand, we want to control the dependence of the constants on the geometry of $K$, through $\rho$ and $\rho _0$; thus, we need the sharper bounds of [20].

In the following, for any $j\in {\mathbb {N}}_0$ and for a Lipschitz open set $D\subset \mathbb {R}^2$, we define the Sobolev seminorms $\left| \phi \right| _{W^{j,\infty }(D)}:=\sup _{{\varvec{\alpha }}\in {\mathbb {N}}^2_0,|{\varvec{\alpha }}|=j}\left\| D^{\varvec{\alpha }}\phi \right\| _{L^\infty (D)}$.

5.1 Exponential Approximation by Circular Waves

The results in Sect. 4 of [20] give the following harmonic approximation estimates.

Proposition 5.1

Under the above assumptions on $\rho ,\rho _0,\eta ,K$, for any real-valued harmonic function $\phi \in W^{1,\infty }(K_\eta )$, there exists a sequence of harmonic polynomials $\{P_N\}_{N\in {\mathbb {N}}_0}$ of degree at most $N$ such that

$$\begin{aligned} \left| \phi -P_N\right| _{W^{j,\infty }(K)}\le C\,h_K^{1-j} e^{-bN} \left\| \nabla \phi \right\| _{L^\infty (K_\eta )} , \end{aligned}$$

(31)

for all $j\in {\mathbb {N}}_0$, where $C>0$ and $b>0$ depend only on $\rho ,\rho _0,\eta$ and $j$. Moreover, $P_N$ interpolates $\phi$ in at least $(N+1)$ points on $\partial K$.

Proof

The proof is a slight improvement of that of Theorem 4.10 and Corollary 4.11 of [20]. Using the same notation of the proof of [20], Theorem 4.10] ($u=\phi$ harmonic to be approximated, $f=u+iv$ holomorphic in ${\mathcal {D}}$), we define $\tilde{f}:=f-u(x_0,y_0)$ and $\tilde{q}_p:=q_p-u(x_0,y_0)$. We have $f-q_p=\tilde{f}-\tilde{q}_p$, $\tilde{u}(x_0,y_0)=v(x_0,y_0)=0$, $\left| \nabla \tilde{u}\right| =\left| \nabla u\right| =\left| \nabla v\right|$ and

$$\begin{aligned} \left\| \tilde{f}\right\| _{L^\infty ({\mathcal {D}})}&\le \left\| \tilde{u}\right\| _{L^\infty ({\mathcal {D}})}+\left\| v\right\| _{L^\infty ({\mathcal {D}})}\\&\le {{\mathrm{diam}}}({\mathcal {D}})\big (\left\| \nabla \tilde{u}\right\| _{L^\infty ({\mathcal {D}})}+\left\| \nabla v\right\| _{L^\infty ({\mathcal {D}})}\big ) =2{{\mathrm{diam}}}({\mathcal {D}})\left\| \nabla u\right\| _{L^\infty ({\mathcal {D}})}, \end{aligned}$$

which shows that the $W^{1,\infty }$-norm at the right-hand side of the bounds in the assertion of [20], Corollary 4.11] can be substituted by the similar seminorm.

The factor $h_K^{j-1}$ follows from a simple affine scaling. $\square$

The explicit values of the constants $C$ and $b$ can easily be computed following the proofs in [20].

In [32], following [40], the $k$-dependent Vekua operators $V_1,V_2:C^0(K)\rightarrow C^0(K)$ were introduced. They are inverses of each other, i.e. they satisfy $V_1=V_2^{-1}$, and are bijective and bicontinuous between the following pairs of spaces (see [32], Theorems 2.5 and 3.1]):

$$\begin{aligned}&{\mathcal H}^j(K):=\big \{\phi \in H^j(K),\, \Delta \phi =0\big \}\\&V_1\downarrow \;\uparrow V_2\\&{\mathcal H}^j_k(K):=\big \{u\in H^j(K),\, \Delta u+k^2u=0\big \} \quad \forall j\in {\mathbb {N}}_0. \end{aligned}$$

In [32], Theorem 3.1], the continuity of these operator in $L^\infty (K)$-norm was also stated. Here, we generalise this result to higher order $W^{j,\infty }(K)$-norms, maintaining an explicit expression of the continuity constants.

Lemma 5.2

For any $j\in {\mathbb {N}}$ and $\phi ,u\in W^{j,\infty }(K)$ such that $\Delta \phi =\Delta u+k^2u=0$ in $K$, we have the continuity bounds:

$$\begin{aligned} \left\| V_1[\phi ]\right\| _{L^{\infty }(K)}&\le \big (1+(kh_K)^2\big )\left\| \phi \right\| _{L^{\infty }(K)}, \end{aligned}$$

(32)

$$\begin{aligned} \left\| V_2[u]\right\| _{L^{\infty }(K)}&\le \bigg (1+\frac{(kh_K)^2 e^{\frac{1}{2}kh_K}}{4}\bigg )\left\| u\right\| _{L^{\infty }(K)}, \end{aligned}$$

(33)

$$\begin{aligned} \left| V_1[\phi ]\right| _{W^{j,\infty }(K)}&\le (1+(kh_K)^2e^j)\left| \phi \right| _{W^{j,\infty }(K)} +k^2h_K e^j\left| \phi \right| _{W^{j-1,\infty }(K)} \end{aligned}$$

(34)

$$\begin{aligned}&\quad +\, (1+j)\big (j+kh_K\big )e^j\sum _{\ell =0}^{j-2} k^{j-\ell }\left| \phi \right| _{W^{\ell ,\infty }(K)},\nonumber \\ \left| V_2[u]\right| _{W^{1,\infty }(K)}&\le k^2h_Ke^{1+\frac{1}{2} kh_K}\left\| u\right\| _{L^{\infty }(K)} +(1+k^2h_K^2e^{\frac{1}{2} kh_K}) \left| u\right| _{W^{1,\infty }(K)}. \end{aligned}$$

(35)

Proof

The two bounds in $L^\infty (K)$-norms are simpler versions of [32], Eqs. (18), (19)]. To prove the remaining ones, we recall that the operators $V_\xi$, with $\xi =1,2$, were defined as $V_\xi [\phi ]({{\mathbf {x}}}):=\phi ({{\mathbf {x}}})+\int _0^1M_\xi ({{\mathbf {x}}},t)\phi (t{{\mathbf {x}}})\,\mathrm {d}t$ for two suitable kernel functions $M_\xi \in C^\infty (K\times [0,1])$ (see [32], Sect. 2]). Thus, using the properties of multi-indices ${\varvec{\alpha }}=(\alpha _1,\alpha _2)\in {\mathbb {N}}^2_0$ and the Leibniz rule for multidimensional derivatives $D^{\varvec{\alpha }}=\frac{\partial ^{|{\varvec{\alpha }}|}}{\partial x_1^{\alpha _1}\partial x_2^{\alpha _2}}$, we have

$$\begin{aligned}&\left| V_\xi [\phi ]\right| _{W^{j,\infty }(K)}= \sup _{{\varvec{\alpha }}\in {\mathbb {N}}_0^2, |{\varvec{\alpha }}|=j}\left\| D^{\varvec{\alpha }}\phi + \int _0^1D^{\varvec{\alpha }}\big (M_\xi ({{\mathbf {x}}},t)\phi (t{{\mathbf {x}}})\big )\,\mathrm {d}t\right\| _{L^{\infty }(K)}\\&\quad =\sup _{{\varvec{\alpha }}\in {\mathbb {N}}^2_0, |{\varvec{\alpha }}|=j}\left\| D^{\varvec{\alpha }}\phi + \int _0^1 \sum _{{\varvec{\beta }}\in {\mathbb {N}}^2_0,{\varvec{\beta }}\le {\varvec{\alpha }}}\left( {\begin{array}{c}{\varvec{\alpha }}\\ {\varvec{\beta }}\end{array}}\right) \big (D^{\varvec{\beta }}M_\xi ({{\mathbf {x}}},t)\big )t^{|{\varvec{\alpha }}-{\varvec{\beta }}|}\big (D^{{\varvec{\alpha }}-{\varvec{\beta }}}\phi \big )|_{t{{\mathbf {x}}}} \big )\,\mathrm {d}t\right\| _{L^{\infty }(K)}\\&\quad \le \left| \phi \right| _{W^{j,\infty }(K)}\!+\! \sup _{{\varvec{\alpha }}\in {\mathbb {N}}^2_0, |{\varvec{\alpha }}|=j}\sum _{{\varvec{\beta }}\in {\mathbb {N}}^2_0,{\varvec{\beta }}\le {\varvec{\alpha }}} \left( {\begin{array}{c}{\varvec{\alpha }}\\ {\varvec{\beta }}\end{array}}\right) \sup _{t\in [0,1]}\left| M_\xi (\cdot ,t)\right| _{W^{|{\varvec{\beta }}|,\infty }(K\times [0,1])} \left| \phi \right| _{W^{|{\varvec{\alpha }}-{\varvec{\beta }}|,\infty }(K)}\\&\quad \le \left| \phi \right| _{W^{j,\infty }(K)}+\sum _{\ell =0}^j (1+\ell )e^j \sup _{t\in [0,1]}\left| M_\xi (\cdot ,t)\right| _{W^{\ell ,\infty }(K\times [0,1])} \left| \phi \right| _{W^{j-\ell ,\infty }(K)}, \end{aligned}$$

where in the last step we used $\left( {\begin{array}{c}{\varvec{\alpha }}\\ {\varvec{\beta }}\end{array}}\right) =\left( {\begin{array}{c}\alpha _1\\ \beta _1\end{array}}\right) \left( {\begin{array}{c}\alpha _2\\ \beta _2\end{array}}\right) \le \frac{\alpha _1^{\beta _1}\alpha _2^{\beta _2}}{\beta _1!\beta _2!}\le e^{|{\varvec{\alpha }}|}$ and the multi-index count [30], Equation (B.10)]. The final estimates follow from the bounds on the kernels $M_\xi$ in [30], Lemma 2.3.3]. $\square$

The results of Lemma 5.2 hold if $K$ is replaced by $K_\eta$ by substituting $h_K$ with $h_K(1+2\eta )$, since $K_\eta$ is star-shaped with respect to the origin.

Following Melenk [25], we say that $u_N\in C^0(K)$ is a generalised harmonic polynomial of degree $N\in {\mathbb {N}}_0$ if its inverse Vekua transform $V_2[u_N]$ is a harmonic polynomial of degree $N$. As described in [30], Sect. 2.4], generalised harmonic polynomials are nothing else than circular waves (often called Fourier–Bessel functions), i.e. smooth solutions of the Helmholtz equation that are separable in polar coordinates: they are linear combinations of

$$\begin{aligned} {{\mathbf {x}}}=(|{{\mathbf {x}}}|\cos \psi ,|{{\mathbf {x}}}|\sin \psi )\quad \longmapsto \quad e^{in\psi }J_{|n|}(k|{{\mathbf {x}}}|), \qquad -N\le n\le N, \end{aligned}$$

where $J_n$ is a Bessel function of the first kind and order $n$.

In the next proposition, we exploit the mapping properties of the Vekua operators proved in Lemma 5.2 to transfer the approximation result for harmonic polynomials and harmonic functions of Proposition 5.1 to generalised harmonic polynomials and Helmholtz solutions (compare with [30], Proposition 3.3.3]).

Proposition 5.3

Under the above assumptions on $\rho ,\rho _0,\eta ,K$, for any $u\in W^{1,\infty }(K_\eta )$ solution of $\Delta u+k^2u=0$, there exists a sequence of generalised harmonic polynomials $\{Q_N\}_{N\in {\mathbb {N}}_0}$ of degree at most $N$ such that

$$\begin{aligned}&\left| u-Q_N\right| _{W^{j,\infty }(K)}\nonumber \\&\quad \le C\,e^{-bN} \, h_K^{1-j} \big (1+(kh_K)^{j+4}\big ) e^{kh_K} \Big (k^2h_K\left\| u\right\| _{L^\infty (K_\eta )}+\left\| \nabla u\right\| _{L^\infty (K_\eta )}\Big ),\quad \end{aligned}$$

(36)

for all $j\in {\mathbb {N}}_0$, where $C>0$ and $b>0$ depend only on $\rho ,\rho _0,\eta$ and $j$.

Proof

For any $N\in {\mathbb {N}}$, define $Q_N=V_1[P_N]$ where $P_N$ is the harmonic polynomial of degree $N$ associated to $V_2[u]$ by Proposition 5.1. Then, for all $j\ge 0$,

$\square$

5.2 Exponential Approximation by Plane Waves

In Proposition 5.4, we prove approximation bounds for plane wave spaces and Helmholtz solutions. The main result is given by the “inversion” of the Jacobi–Anger expansion obtained in [30], Lemma 3.4.3]; this allows to approximate circular waves with plane waves with more than exponential convergence in the number of plane waves. The final bound is then obtained with a triangular inequality argument, Cauchy’s estimates for Helmholtz solutions and Proposition 5.3.

The whole proof is just a modification of those in Sect. 3.4.2 and 3.5 of [30] (see in particular Remark 3.5.8 therein). The main differences are as follows: (i) here, we never use $H^{j}$-type Sobolev norms but only $W^{j,\infty }$-type, (ii) we aim for exponential convergence and require that the function to be approximated be defined in a neighbourhood of the element, and (iii) the bounds coming from [20] allow to reduce the dependence of the bounding constant on the element shape to the parameters $\rho$ and $\rho _0$ only.

Proposition 5.4

Fix $q\in {\mathbb {N}}$ and $p=2q+1$ different unit vectors (the propagation directions) $\{{{\mathbf {d}}}_m=(\cos \theta _m,\sin \theta _m)\}_{m=-q}^q$. Assume there exists $0<\zeta \le 1$ such that

$$\begin{aligned} \min _{\begin{array}{c} m,m'=-q,\ldots ,q\\ m\ne m' \end{array}}\big |\theta _m-\theta _{m'}\big |\ge \frac{2\pi }{p}\;\zeta . \end{aligned}$$

(37)

Fix $u\in W^{1,\infty }(K_\eta )$ solution of $\Delta u+k^2u=0$. Then, under the above assumptions on $\rho$, $\rho _0$, $\eta$, $K$, there exists a linear combination of plane waves with propagation directions $\{{{\mathbf {d}}}_m\}_{m=-q}^q$ which approximates $u$ with the following error bound:

$$\begin{aligned}&\left| u-\sum _{m=-q}^q\alpha _m e^{ik{{\mathbf {x}}}\cdot {{\mathbf {d}}}_m}\right| _{W^{j,\infty }(K)}\\&\le C\big (1+(kh_K)^{j+7}\big )e^{3kh_K} h_K^{-j} \Bigg [(kh_K)e^{-bq} + \frac{h_K(kh_K)^{\lfloor q/2\rfloor +1}\big (1+(kh_K)^{\lfloor \frac{q-1}{2}\rfloor }\big )}{\big (c_0\zeta ^4(q+1)\big )^{\frac{q}{2}}}\Bigg ]\\&\qquad \qquad \cdot \Big (k\left\| u\right\| _{L^\infty (K_\eta )}+\left\| \nabla u\right\| _{L^\infty (K_\eta )}\Big ) \end{aligned}$$

for all $j\in {\mathbb {N}}_0$, where $C>0$ and $b>0$ depend only on $\rho ,\rho _0,\eta$ and $j$, while $c_0>0$ is independent of all the other parameters.

Proof

We consider $N\in {\mathbb {N}}$ such that $N\le \lfloor (q-1)/2\rfloor$ and, using plane waves, we approximate the circular wave $Q_N$ given by Proposition 5.3.

First we note that Vekua’s theory allows to extend Cauchy’s estimates for harmonic functions to Helmholtz solutions. In particular, we can control the $W^{j,\infty }(K)$-norm at the left-hand side in the assertion’s bound with the $L^{\infty }(K_{\eta })$-norm of the same function: for any $w\in L^\infty (K_\eta )$, $\Delta w+k^2 w=0$,

(38)

where the constant $C$ depends only on $j$.

We obtain the order of convergence of the plane wave approximation of $Q_N$ from Lemma 3.4.3 of [30] (together with $K_\eta \subset B_{(1-\rho +\eta )h_K}$, $\left\| \cdot \right\| _{L^2(K)}\le h_K\left\| \cdot \right\| _{L^\infty (K)}$, and setting $K=0$ in the notation of [30], Lemma 3.4.3]): there exists $\mathbf {\alpha }\in \mathbb {C}^p$ such that

$$\begin{aligned}&\left\| Q_N-\sum _{m=-q}^q\alpha _m e^{ik{{\mathbf {x}}}\cdot {{\mathbf {d}}}_m}\right\| _{L^{\infty }(K_\eta )} \le \left\| Q_N-\sum _{m=-q}^q\alpha _m e^{ik{{\mathbf {x}}}\cdot {{\mathbf {d}}}_m}\right\| _{L^{\infty }(B_{(1-\rho +\eta )h_K})}\nonumber \\&\quad \le \frac{e^3}{\pi ^\frac{3}{2}\rho ^{N+1}} \left( \frac{e^{\frac{5}{2}}}{2\sqrt{2}\;\zeta ^2}\right) ^q \left( 2^N\sqrt{N+1}\right) \left( 1+(k h_K)^{-N}\right) \nonumber \\&\qquad \cdot e^{\frac{(1-\rho +\eta )kh_K}{2}} \frac{\big ((1-\rho +\eta )kh_K\big )^{q+1}}{(q+1)^{\frac{q+1}{2}}}\left\| V_2[Q_N]\right\| _{L^\infty (K)} \end{aligned}$$

(39)

The norm of the harmonic polynomial $V_2[Q_N]$ is immediately controlled by that of $u$ using the triangle inequality and recalling the definition of $Q_N$:

$$\begin{aligned} \left\| V_2[Q_N]\right\| _{L^\infty (K)}\le & {} \left\| V_2[u]\right\| _{L^\infty (K)}+\left\| V_2[u]-V_2[Q_n]\right\| _{L^\infty (K)}\nonumber \\&\overset{(31)}{\le }C\Big (\left\| V_2[u]\right\| _{L^\infty (K_\eta )}+h_K\left\| \nabla V_2[u]\right\| _{L^\infty (K_\eta )}\Big ) \nonumber \\&\overset{(33),(35)}{\le }C\big (1+(kh_K)^2\big )e^{\frac{1}{2}(1+2\eta )kh_K} \Big (\left\| u\right\| _{L^\infty (K_\eta )}+h_K\left\| \nabla u\right\| _{L^\infty (K_\eta )}\Big )\nonumber \\ \end{aligned}$$

(40)

where $C>0$ only depends on $\rho$, $\rho _0$ and $\eta$.

We now put together the various bounds: the plane wave approximation error is split using the triangle inequality in a Fourier–Bessel approximation error (controlled in Proposition 5.3) and in a remainder term controlled by (39) (using (38) to reduce the order of the norm):

$$\begin{aligned}&\left| u-\sum _{m=-q}^q\alpha _m e^{ik{{\mathbf {x}}}\cdot {{\mathbf {d}}}_m}\right| _{W^{j,\infty }(K)} \le \left| u-Q_N\right| _{W^{j,\infty }(K)} +\left| Q_N-\sum _{m=-q}^q\alpha _m e^{ik{{\mathbf {x}}}\cdot {{\mathbf {d}}}_m}\right| _{W^{j,\infty }(K)}\\&\quad \overset{(38)}{\le } \left| u-Q_N\right| _{W^{j,\infty }(K)} \!+\!C \big (1 +(kh_K)^{j+4}e^{{\frac{1}{2}}kh_K}\big )h_K^{-j}\left\| Q_N\!-\!\sum _{m=-q}^q\alpha _m e^{ik{{\mathbf {x}}}\cdot {{\mathbf {d}}}_m}\right\| _{L^{\infty }(K_\eta )}\\&\quad \overset{(36),(39)}{\le }C e^{-bN} h_K^{1-j} \big (1+(kh_K)^{j+4}\big ) e^{kh_K} \Big (k^2h_K\left\| u\right\| _{L^\infty (K_\eta )}+\left\| \nabla u\right\| _{L^\infty (K_\eta )}\Big )\\&\qquad +C\big ((kh_K)^{-N}\!+\!(kh_K)^{j+4}\big )h_K^{-j} \left( \frac{3e^{\frac{5}{2}}}{4\sqrt{2}\;\zeta ^2}\right) ^q \left( 2^N\sqrt{N\!+\!1}\right) e^{{\frac{5}{4}}kh_K}\\&\qquad \qquad \cdot \frac{(kh_K)^{q+1}}{(q+1)^{\frac{q+1}{2}}} \left\| V_2[Q_N]\right\| _{L^\infty (K)}\\&\quad \overset{(40)}{\le }C \big (1+(kh_K)^{j+7}\big )e^{\frac{9}{4}kh_K} h_K^{-j} \\&\qquad \qquad \cdot \Bigg [(kh_K)e^{-bN} + \big (1+(kh_K)^{-N}\big ) \left( \frac{3e^{\frac{5}{2}}}{4\sqrt{2}\;\zeta ^2}\right) ^q \left( 2^N\sqrt{N+1}\right) \frac{h_K\,(kh_K)^{ q}}{(q+1)^{\frac{q+1}{2}}}\Bigg ]\\&\qquad \qquad \cdot \Big (k\left\| u\right\| _{L^\infty (K_\eta )}+\left\| \nabla u\right\| _{L^\infty (K_\eta )}\Big ), \end{aligned}$$

where $C$ and $b$ depend on $j, \rho ,\rho _0,\eta$ only. We now fix $N:=\lfloor \frac{q-1}{2}\rfloor$ and obtain the assertion (with $c_0>0.0119$)

$$\begin{aligned} \left| u-\sum _{m=-q}^q\alpha _m e^{ik{{\mathbf {x}}}\cdot {{\mathbf {d}}}_m}\right| _{W^{j,\infty }(K)} \le&\,C\big (1+(kh_K)^{j+ 7}\big )e^{3kh_K} h_K^{-j} \left[ (kh_K)e^{-\frac{b}{2}q} \right. \\&\left. \quad + \left( \frac{3e^{\frac{5}{2}}}{4\zeta ^2}\right) ^q \frac{h_K(kh_K)^{\lfloor q/2\rfloor +1}\big (1+(kh_K)^{\lfloor \frac{q-1}{2}\rfloor }\big )}{(q+1)^{\frac{q}{2}}}\right] \\&\qquad \cdot \Big (k\left\| u\right\| _{L^\infty (K_\eta )}+\left\| \nabla u\right\| _{L^\infty (K_\eta )}\Big ). \end{aligned}$$

$\square$

Remark 5.5

If $kh_K\gg 1$, the numerator of the fraction in the bound in Proposition 5.4 behaves like $2h_K(kh_K)^q$ and can badly affect the convergence of the approximation by generating a long pre-asymptotic regime in $q$ (compare with the “step” in Fig. 3.1 of [17]). This term comes from bound (3.42) in [30], which can be improved to

$$\begin{aligned} \sup _{t_\in [0,kh_K]}\sum _{\ell >q}|J_\ell (t)| \le \sum _{\ell >q}\frac{(kh_K/2)^\ell }{\ell !} \le \frac{e^{kh_K/2}\;\gamma (q+1,kh_K/2)}{q!}, \end{aligned}$$

where $\gamma (a,x):=\int _0^xe^{-t}t^{a-1}\,\mathrm {d}t=\Gamma (a)x^ae^{-x}\sum _{n\ge 0}\frac{x^n}{\Gamma (a+n+1)}$ is the lower incomplete gamma function [1], 6.5.2, 6.5.4, 6.5.29]. Using this, the numerator can be reduced to $h_K q 2^{q+1}\gamma (q,kh_K/2)$, which has similar behaviour to $2h_K(kh_K)^q$ for large values of $kh_K$ and $q$, but is considerably smaller.

6 Exponential Convergence

As in the case of standard polynomial finite elements, we establish exponential convergence of $\left\| u-u_{hp}\right\| _{0,\Omega }$ in terms of the number of degrees of freedom for particular families of meshes.

6.1 Geometric Meshes

We restrict ourselves to special instances of families of meshes given by sequences $\{{\mathcal T}_{L}\}_{L\in {\mathbb {N}}}$ of so-called geometrically graded meshes indexed by a refinement level $L$ denoting the number of element layers in the mesh, see Assumption 6.1 below. Meshes of this type with simple polygonal or polyhedral elements have universally been used for conventional $hp$-finite element methods [38]. Conversely, we demand only compliance of $\{{\mathcal T}_{L}\}_{L\in {\mathbb {N}}}$ with Assumptions (M1) and (M2) from Sect. 4.1, and, thus, rather general shapes of the elements are admitted. We impose the following properties on admissible geometrically graded meshes.

Assumption 6.1

Let $0<\sigma <1$ be a fixed grading parameter. The elements of every mesh ${\mathcal T}_{L}$, $L\in {\mathbb {N}}$, can be grouped into layers ${\mathcal L}^{L}_{\ell }$, $0\le \ell \le L$, that is,

$$\begin{aligned} {\mathcal T}_{L} = \bigcup \limits _{\ell =0}^{L}{\mathcal L}_{\ell }^{L},\qquad {\mathcal L}_{\ell }^{L}\cap {\mathcal L}_{\ell '}^{L}=\emptyset \,\text { if }\,\ell \not =\ell ', \end{aligned}$$

such that:

(GM1):

the $L$th layer ${\mathcal L}_{L}^{L}$ contains the set of elements abutting a corner;

(GM2) :

except for the elements in ${\mathcal L}_{L}^{L}$, the distance of an element from the nearest corner point depends geometrically on its layer index (recalling that ${\mathcal {C}}= \left\{ {{\mathbf {c}}}_{\nu }\right\} _{\nu =1}^{n_c}$ is the set of corner points):

$$\begin{aligned}&\exists C>0:\quad C^{-1}\sigma ^{\ell } \le {\text {dist}}(K,{\mathcal {C}}) \le C \sigma ^{\ell }\nonumber \\&\quad \forall K\in {\mathcal L}^{L}_{\ell },\quad 0\le \ell \le L-1, \quad L \in {\mathbb {N}}; \end{aligned}$$

(41)

(GM3) :

the size of an element depends geometrically on its layer index: $\exists C>0$ such that

$$\begin{aligned} \begin{array}{cccc} &{}C^{-1}\le h_{K} \le C&{}\forall K\in {\mathcal L}^{L}_{0},&{}\quad L \in {\mathbb {N}},\\ &{}C^{-1}\sigma ^{\ell -1}(1-\sigma ) \le h_{K} \le C \sigma ^{\ell -1}(1-\sigma )&{}\quad \forall K\in {\mathcal L}^{L}_{\ell },&{}\quad 1\le \ell \le L-1, \\ &{}C^{-1}\sigma ^{L-1}\le h_{K} \le C \sigma ^{L-1} &{}\forall K\in {\mathcal L}^{L}_{L};&{} \end{array}\qquad \end{aligned}$$

(42)

(GM4) :

for $\ell \ge 2$, ${\mathcal T}_{L}$ is obtained from ${\mathcal T}_{L-1}$ by refining only elements of ${\mathcal L}_{L-1}^{L-1}$ (i.e. ${\mathcal L}^L_\ell ={\mathcal L}^{L'}_\ell$ for all $\ell <\min \{L,L'\}$).

Here and in the sequel, we adhere to the convention that a “generic constant” $C>0$ must depend neither on refinement levels $\ell$ and $L$, nor on the grading parameter $\sigma$, nor on the solution $u$.

We remind that (GM2) and (GM3) imply that the diameter of an element in the $\ell$th layer is proportional to its distance from the nearest corner:

$$\begin{aligned} \begin{aligned} \exists C>0:\qquad \frac{1}{C} {\text {dist}}(K,{\mathcal {C}})&\le h_{K} \le C {\text {dist}}(K,{\mathcal {C}}) \qquad&\forall K\in {\mathcal L}_{0}^{L}, L\in {\mathbb {N}},\\ \frac{1}{C}\,\frac{1-\sigma }{\sigma } {\text {dist}}(K,{\mathcal {C}})&\le h_{K} \le C\, \frac{1-\sigma }{\sigma } {\text {dist}}(K,{\mathcal {C}}) \quad&\forall K\in {\mathcal L}_{\ell }^{L}, 1\le \ell \le L-1. \end{aligned} \end{aligned}$$

(43)

Appealing to (M1) and (GM3), we can control the area of the elements in a particular layer:

$$\begin{aligned} \begin{aligned} \exists C>0:\qquad \qquad \qquad \frac{1}{C}&\le |K| \le h_{K}^{2} \le C \qquad&\forall K\in {\mathcal L}_{0}^{L}, L\in {\mathbb {N}},\\ \frac{1}{C} \sigma ^{2(\ell -1)}(1-\sigma )^2&\le |K| \le h_{K}^{2} \le C \sigma ^{2(\ell -1)}(1-\sigma )^2 \quad&\forall K\in {\mathcal L}_{\ell }^{L}, 1\le \ell \le L-1,\\ \frac{1}{C} \sigma ^{2(L-1)}&\le |K| \le h_{K}^{2} \le C \sigma ^{2(L-1)} \qquad&\forall K\in {\mathcal L}_{L}^{L}. \end{aligned} \end{aligned}$$

As a consequence of the mesh construction, the area occupied by the $\ell$th layer is bounded as follows:

$$\begin{aligned} \begin{aligned}&\text {area}({\mathcal L}_{0}^{L})\le C, \qquad \text {area}({\mathcal L}_{L}^{L})\le C\sigma ^{2(L-1)}\\&\text {area}({\mathcal L}_{\ell }^{L})\!=\!\text {area}({\mathcal L}_{\ell }^{\ell +1})\!=\!\text {area}({\mathcal L}_{\ell }^{\ell }) -\text {area}({\mathcal L}_{\ell +1}^{\ell +1})\le C\sigma ^{2\ell }\,\frac{1-\sigma ^2}{\sigma ^2},\quad 1\le \ell \le L-1. \end{aligned} \end{aligned}$$

Taking the ratio of the areas in the last two formulae, we thus conclude that the number of elements per layer is uniformly bounded in $\ell$:

$$\begin{aligned} \begin{aligned} \exists C>0:\qquad \qquad \sharp {\mathcal L}_{\ell }^{L}&\le C\,\frac{1+\sigma }{1-\sigma }, \qquad \qquad 1\le \ell \le L-1,\\ \sharp {\mathcal L}_{L}^{L}, \sharp {\mathcal L}_{0}^{L}&\le C, \quad L\in {\mathbb {N}}. \end{aligned} \end{aligned}$$

(44)

Immediate from (44) is the fact that geometrically graded meshes satisfy (M3) because, retaining the notation ${\mathcal F}_h^I$ for the set of interior edges of some ${\mathcal T}_{L}$,

$$\begin{aligned} \left| {\mathcal F}_h^I\right| \overset{\text {}(21)}{\le } C \sum \limits _{K\in {\mathcal T}_{L}} h_{K}&\le C \left[ \sum \limits _{K\in {\mathcal L}_{0}^{L}} h_K +\sum \limits _{K\in {\mathcal L}_{L}^{L}} h_K +\sum \limits _{\ell =1}^{L-1}\sum \limits _{K\in {\mathcal L}_{\ell }^{L}} h_{K} \right] \nonumber \\&\overset{\hbox {(GM3)},(44)}{\le } C \left[ 1+\sigma ^{L-1}+\sum \limits _{\ell =1}^{L-1}\frac{1+\sigma }{1-\sigma }\, \sigma ^{\ell -1}(1-\sigma ) \right] \nonumber \\&=C\left[ 1+\sigma ^{L-1}+(1+\sigma )\,\frac{1-\sigma ^{L-1}}{1-\sigma } \right] \overset{{0<\sigma <1}}{\le }C\,\frac{1}{1-\sigma }=:C_{\mathcal F},\nonumber \\ \end{aligned}$$

(45)

with all constants independent of $L$.

6.2 Plane Wave $hp$-Spaces

The gist of $hp$-approximation is to raise the number of plane waves used on each element along with refining the mesh. This is reflected in the construction of the plane wave $hp$-spaces based on a sequence of geometrically graded meshes $\left\{ {\mathcal T}_{L}\right\} _{L\in {\mathbb {N}}}$ as introduced in Sect. 6.1. To begin with, we set the dimension of the local plane wave spaces to

$$\begin{aligned} p(L) := 2L +1,\quad L\in {\mathbb {N}}. \end{aligned}$$

(46)

For the sake of simplicity, we opt for equispaced plane wave directions (i.e. $\zeta =1$ in Proposition 5.4)

$$\begin{aligned} {{\mathbf {d}}}_{m}^{p} = \left( {\begin{array}{c}\cos (\frac{2\pi }{p}m)\\ \sin (\frac{2\pi }{p}m)\end{array}}\right) ,\quad 0\le m < p,\quad p\in {\mathbb {N}}, \end{aligned}$$

which give rise to the local plane wave spaces

$$\begin{aligned} P\!W_{p,k}{(K)} {:=} \bigg \{v\in C^{\infty }(\mathbb {R}^{2}): v({{\mathbf {x}}}) {=}\! \sum \limits _{m=0}^{p-1} \alpha _{m} \exp \left( ik{{\mathbf {d}}}_{m}^{p}\cdot {({{\mathbf {x}}}\!-\!{{\mathbf {x}}}_K)}\right) ,\,\alpha _{m}\in \mathbb {C}\bigg \}, p\in {\mathbb {N}}. \end{aligned}$$

where ${{\mathbf {x}}}_K$ was defined in Assumption (M1), Sect. 4.1. Then, the trial and test spaces for the $hp$-version of the Trefftz-DG method of Sect. 3 are defined as

$$\begin{aligned} V_{L} := \big \{v\in L^{2}(\Omega ):\; \left. v\right. _{|K} \in P\!W_{p(L),k}{(K)}\; \forall K\in {\mathcal T}_{L}\big \}, \end{aligned}$$

and the corresponding solution will be denoted by $u_{L}\in V_{L}$. Obviously, thanks to the bound on the number of elements per layer (44), the total number of degrees of freedom, which is $\dim V_{L}$, is bounded by

$$\begin{aligned} \dim V_{L} \le C\frac{1}{1-\sigma } L\,p(L)\quad \forall L\in {\mathbb {N}}. \end{aligned}$$

(47)

According to Theorem 4.6 and the bound on $|{\mathcal F}^I_h|$ (45), an $L$-uniform bound of the discretisation error $\left\| u-u_{L}\right\| _{0,\Omega }$ is provided by $\left\| u-v_{L}\right\| _{DG^{+}}$ for any $v_{L}\in V_{L}$. A concrete choice of $v_{L}$ will rely on particular local approximations of $u$ chosen differently for elements away from corners, see Sect. 6.3, and elements at corners, see Sect. 6.4.

Before we give details, we elaborate a simpler bound for $\left\| u-v_{L}\right\| _{DG^{+}}$. Immediate from the definition of $\left\| \cdot \right\| _{DG^{+}}$ is

$$\begin{aligned} \left\| u-v_L\right\| _{DG^+}^2\le&\,C \sum _{K\in {\mathcal T}_L}\bigg ( k^{-1}\left\| \beta ^{1/2}\nabla (u-v_L)\cdot {{\mathbf {n}}}\right\| _{0,\partial K\setminus \partial \Omega }^2 \\&+k\left\| \alpha ^{1/2}(u-v_L)\right\| _{0,\partial K\setminus \Gamma _R}^2 +k\left\| \beta ^{-1/2}(u-v_L)\right\| _{0,\partial K\setminus \partial \Omega }^2\\&+k^{-1}\left\| \alpha ^{-1/2}\nabla (u-v_L)\cdot {{\mathbf {n}}}\right\| _{0,\partial K\setminus \Gamma _R}^2\\&+ k^{-1}\left\| \delta ^{1/2}\vartheta ^{-1/2} \nabla (u-v_L)\cdot {{\mathbf {n}}}\right\| _{0,\partial K\cap \Gamma _R}^2\\&+k\left\| (1-\delta )^{1/2}\vartheta ^{1/2}(u-v_L)\right\| _{0,\partial K\cap \Gamma _R}^2\\&+k\left\| \delta ^{-1/2}\vartheta ^{1/2}(u-v_L)\right\| _{0,\partial K\cap \Gamma _R}^2 \bigg ). \end{aligned}$$

Thanks to the particular choice of the parameters $\alpha$, $\beta$ and $\delta$ made in (17) and (18), we thus arrive at the bound

$$\begin{aligned} \left\| u-v_L\right\| _{DG^+}^2\le \,C \sum _{K\in {\mathcal T}_{L}}\Big ( k^{-1}\left\| \nabla (u-v_L)\cdot {{\mathbf {n}}}\right\| _{0,\partial K}^2+ \frac{kh_{\max }}{h_K}\left\| u-v_L\right\| _{0,\partial K}^2 \Big ),\nonumber \\ \end{aligned}$$

(48)

where we have used the fact that the local quasi-uniformity Assumption (M2) implies $h_{f} \le \tau h_{K}$ for any face $f$ of the element $K$; thus, in the estimate (48), $C$ depends on the local quasi-uniformity of the mesh.

6.3 Estimates Away from Corners

A simple consequence of Theorem 2.3 is the possibility to extend $u$ analytically beyond $\partial K$, provided that $K$ does not abut a corner. The solution can be extended to a distance from $K$ proportional to the distance from the closest domain corner, thus proportional to the diameter of $K$ itself, thanks to relation (43). The proof is similar to that of [20], Lemma 5.4] and given for convenience.

Lemma 6.2

There exists $\eta _{*}>0$ depending only on the shape of $\Omega$ and on $\sigma$, in particular, independent of $u$, $k$ and $L\in {\mathbb {N}}$, such that the solution $u$ of (1) is analytic in

$$\begin{aligned} K_{\eta _{*}} := \big \{{{\mathbf {x}}}\in \mathbb {R}^{2}:\; {\text {dist}}({{\mathbf {x}}},K) < \eta _{*}h_{K}\big \}, \end{aligned}$$

and belongs to $W^{1,\infty }(K_{\eta _*})$ for all $K\in {\mathcal T}_{L}\setminus {\mathcal L}_{L}^{L}$, that is, for all elements not adjacent to a corner.

Proof

It goes without saying that we will rely on (11) from Theorem 2.3. For ${{\mathbf {x}}}\in \Omega$, by the geometric triangle inequality, we have the simple estimate

$$\begin{aligned} \widehat{\Phi }({{\mathbf {x}}}) = \prod \limits _{\nu =1}^{n_{c}}\left| {{\mathbf {x}}}-{{\mathbf {c}}}_{\nu }\right| \ge |{{\mathbf {x}}}-{{\mathbf {c}}}_{\mu }| \;2^{1-n_{c}} \prod \limits _{\begin{array}{c} \nu =1\\ \nu \not =\mu \end{array}}^{n_{c}} |{{\mathbf {c}}}_{\mu }-{{\mathbf {c}}}_{\nu }|, \end{aligned}$$

if $\mu$ is the index of the corner closest to ${{\mathbf {x}}}$. Hence, for ${{\mathbf {x}}}\in K$, $K\in {\mathcal T}_{L}\setminus {\mathcal L}_{L}^{L}$, we find the lower bound

$$\begin{aligned} \widehat{\Phi }({{\mathbf {x}}}) \ge C_{\mathcal {C}}\,{\text {dist}}(K,\mathcal {C}),\qquad C_{\mathcal {C}} := 2^{1-n_{c}} \min \limits _{\mu \in \{1,\ldots ,n_{c}\}} \prod \limits _{\begin{array}{c} \nu =1\\ \nu \not =\mu \end{array}}^{n_{c}}|{{\mathbf {c}}}_{\mu }-{{\mathbf {c}}}_{\nu }|. \end{aligned}$$

Thus, from (11), we conclude that $u$ is analytic in

$$\begin{aligned}&\bigcup \limits _{{{\mathbf {x}}}_{0}\in K} \left\{ {{\mathbf {x}}}\in \mathbb {R}^{2}:\,|{{\mathbf {x}}}-{{\mathbf {x}}}_{0}| < \frac{C_{\mathcal {C}} {\text {dist}}(K,\mathcal {C})}{4e\gamma } \right\} \\&\qquad = \left\{ {{\mathbf {x}}}\in \mathbb {R}^{2}:\, {\text {dist}}({{\mathbf {x}}},K) < \frac{C_{\mathcal {C}}}{4e\gamma }{\text {dist}}(K,\mathcal {C}) \right\} . \end{aligned}$$

The distance ${\text {dist}}(K,\mathcal {C})$ is related to the size of $K$ by (43), which provides $C^{-1} \frac{\sigma }{1-\sigma }h_{K} \le {\text {dist}}(K,\mathcal {C})$, if $K\in {\mathcal L}_{\ell }^{L}$, $1\le \ell \le L-1$, or $C^{-1} h_{K} \le {\text {dist}}(K,\mathcal {C})$, if $K\in {\mathcal L}_{0}^{L}$, where the constant $C$ is that in (43). This yields the assertion of the lemma, for instance, for the choice $\eta _{*} =\min \{1,\frac{\sigma }{1-\sigma }\}\frac{C_{\mathcal {C}}}{8e\gamma C}$. $\square$

From this lemma, it is immediate that $u \in L^\infty (K)$ and $\nabla u\in L^\infty (K)^2$ for every element $K\in {\mathcal T}_{L}\setminus {\mathcal L}_{L}^{L}$. Now we fix such an element $K$. If $w\in L^\infty (K)$, the consequence (21) of the star-shapedness of $K$ gives

$$\begin{aligned} \left\| w\right\| _{0,\partial K}^2\le \left| \partial K\right| \left\| w\right\| _{L^\infty (K)}^2 \le C\, h_K\left\| w\right\| _{L^\infty (K)}^2. \end{aligned}$$

Hence, the contribution of the elements $K\in {\mathcal T}_{L}\setminus {\mathcal L}_{L}^{L}$ to the right-hand side of estimate (48) can be bounded by

$$\begin{aligned}&\sum _{K\in {\mathcal T}_{L}\setminus {\mathcal L}_{L}^{L}}\Big ( k^{-1}\left\| \nabla (u-v_L)\cdot {{\mathbf {n}}}\right\| _{0,\partial K}^2+ \frac{kh_{\max }}{h_K}\left\| u-v_L\right\| _{0,\partial K}^2 \Big ) \nonumber \\&\qquad \le \sum _{K\in {\mathcal T}_{L}\setminus {\mathcal L}_{L}^{L}}C\left( \frac{h_K}{k}\left\| \nabla (u-v_L)\right\| _{L^\infty (K)}^2+ {kh_{\max }}\left\| u-v_L\right\| _{L^\infty (K)}^2\right) . \end{aligned}$$

(49)

Along with Lemma 6.2, this paves the way for using the approximation result of Proposition 5.4 for $\zeta =1$ [defined in (37)] locally on each element $K\in {\mathcal L}_{\ell }^{L}$, $0\le \ell \le L-1$: picking $v_{L}\in PW_{p(L),k}(K)$ as a suitable linear combination of equispaced plane waves according to Proposition 5.4, we find

$$\begin{aligned}&\frac{h_K}{k}\left\| \nabla (u-v_L)\right\| _{L^\infty (K)}^{2} + {kh_{\max }}\left\| u-v_L\right\| _{L^\infty (K)}^{2}\nonumber \\&\quad \!\le \! C\, (kh_{\max })^{16}e^{6kh_K}\, (kh_K)^{-1} \left[ (kh_K)e^{-bL} \!+\! \frac{h_K(kh_K)^{\lfloor L/2\rfloor \!+\!1}\big (1\!+\!(kh_K)^{\lfloor \frac{L-1}{2}\rfloor }\big )}{\big (c_0(L+1)\big )^{\frac{L}{2}}}\right] ^2 \nonumber \\&\qquad \cdot \Big (k\left\| u\right\| _{L^\infty (K_{\eta _*})}+\left\| \nabla u\right\| _{L^\infty (K_{\eta _*})}\Big )^2, \nonumber \\&\quad \le C\, (kh_{\max })^{17}e^{6kh_K}\, h_K \left[ e^{-bL} + \frac{h_K\big (1+(kh_K)^{L-1}\big )}{\big (c_0(L+1)\big )^{\frac{L}{2}}}\right] ^2\nonumber \\&\qquad \cdot \Big (k\left\| u\right\| _{L^\infty (K_{\eta _*})}+\left\| \nabla u\right\| _{L^\infty (K_{\eta _*})}\Big )^2. \end{aligned}$$

(50)

The constant $C$ essentially agrees with the constant $C$ in the assertion of Proposition 5.4 and inherits its dependency on $\rho$, $\rho _0$ and $\eta _*$. The exponential rate $b$ is the same as in Propositions 5.1 and 5.4.

6.4 Estimates at Corners

On $K\in {\mathcal L}_{L}^{L}$, we can neither take for granted $\nabla u\in L^\infty (K)$, nor analyticity of $u$ beyond $\partial K$. Fortunately, since the combined area of these elements is very small for large $L$, simple local estimates suffice. Our aim is to control the terms relative to $K$ in (48) with some bounded function of $u$, independent of $K$, multiplied with any positive powers of $h_K$; then, the geometric scaling (42) for $\ell =L$ provides exponential convergence in $L$.

The first tool we need is the polynomial quasi-interpolation operators $Q^{m}$, $m=1,2$, introduced in [5], Chapter 4], which project onto the spaces $\mathbb {P}_{m-1}$ of two-variate polynomials of degree at most $m-1$. In particular, we make use of $Q_{\hat{K}}^{1}$ and $Q_{\hat{K}}^{2}$ for each $\hat{K}$, where $\hat{K}$ is the scaling of the element $K\in {\mathcal T}_L^L$ as introduced in Sect. 4.1. We remind that the projectors $Q^{m}$ rely on Taylor expansions averaged over $B_{\rho _{0}}$. Then [5], Corollary (4.1.15)] gives us

$$\begin{aligned} \left| Q_{\hat{K}}^{m}\hat{w}\right| _{j,B_{1}} \le C_{m,j} \left\| \hat{w}\right\| _{0,B_{\rho _{0}}} \quad {\forall \hat{w}\in H^{m-1}(\hat{K})},\quad j=0,1,\quad m=1,2, \end{aligned}$$

(51)

with constants $C_{m,j}$ depending only on $\rho _{0}$. Moreover, by the Bramble–Hilbert Lemma from [5], Lemma (4.3.8)], we know

$$\begin{aligned} \left\| \hat{w}-Q_{\hat{K}}^{m}\hat{w}\right\| _{0,\hat{K}} \le C_{m} \left| \hat{w}\right| _{m,\hat{K}}\quad {\forall \hat{w}\in H^{m}(\hat{K})},\quad m=1,2, \end{aligned}$$

(52)

where $C_{m}$ depends on $\rho _{0}$ only. By interpolation between $H^{2}(\hat{K})$ and $L^{2}(\hat{K})$ of the operator $(\mathrm {Id}-Q_{\hat{K}}^m)$ taking values in $L^2(\hat{K})$, we conclude from (51) and (52) for $m=2$ and $j=0$

$$\begin{aligned} \left\| \hat{w}-Q_{\hat{K}}^{2}\hat{w}\right\| _{0,\hat{K}} \le C \left| \hat{w}\right| _{\frac{3}{2}+ s,\hat{K}}\quad {\forall \hat{w}\in H^{\frac{3}{2}+ s}(\hat{K})}, \quad s\in (0,1/2), \end{aligned}$$

(53)

with, as before, $C$ depending on $\rho _{0}$ only. Next [5], Lemma (4.1.17)] asserts that $\nabla \circ Q^{2}_{\hat{K}} = Q^{1}_{\hat{K}}\circ \nabla$, which yields, by interpolation between $H^{1}(\hat{K})$ and $L^{2}(\hat{K})$, applying (51) and (52) to $\nabla \hat{w}$ with $m=1$ and $j=0$,

$$\begin{aligned} \left\| \nabla (\hat{w}-Q^{2}_{\hat{K}}\hat{w})\right\| _{0,\hat{K}} \le C \left| \nabla \hat{w}\right| _{\frac{1}{2}+ s,\hat{K}}\quad {\forall \hat{w}\in H^{\frac{3}{2}+ s}(\hat{K})}, \quad s\in (0,1/2). \end{aligned}$$

(54)

The second tool is a set of special results about the approximation of polynomials by plane waves which can be derived combining Lemma 3.10 and Proposition 3.9 in [13]. In that article, the estimates target a family of triangles and the unit square, here we need the estimates on the unit disc only.

Lemma 6.3

For odd $p\ge 5$, $\hat{k}>0$, and any $\hat{p}_{1} \in \mathbb {P}_{1}(B_{1})$, we can find $\hat{v}_{p}\in P\!W_{p,\hat{k}}(B_{1})$ such that

$$\begin{aligned} \left\| \hat{p}_1-\hat{v}_p\right\| _{0,B_{1}}&\le C\hat{k}^2\left\| \hat{p}_1\right\| _{0,B_{1}}, \end{aligned}$$

(55)

$$\begin{aligned} \left| \hat{p}_1-\hat{v}_p\right| _{1,B_{1}}&\le C(\hat{k}+1)\hat{k}^2\left\| \hat{p}_1\right\| _{0,B_{1}}, \end{aligned}$$

(56)

$$\begin{aligned} \left| \hat{v}_p\right| _{2,B_{1}}&\le C (\hat{k}+1)^2\hat{k}^2\left\| \hat{p}_1\right\| _{0,B_{1}}. \end{aligned}$$

(57)

Based on this lemma, we prove other auxiliary estimates.

Lemma 6.4

Fix odd $p\ge 5$ and $s\in (0,1/2)$. For every $K\in {\mathcal T}_{L}$ and $u\in H^{\frac{3}{2}+s}(K)$, we can find $v_{p}\in P\!W_{p,k}(K)$ such that

$$\begin{aligned} \left\| u-v_p\right\| _{0,K}^2&\le C \left( h_K^{3+2s} \left| u\right| _{\frac{3}{2}+s,K}^2 +h_K^{4}k^4\left\| u\right\| _{0,K}^2\right) , \end{aligned}$$

(58)

$$\begin{aligned} \left| u-v_p\right| _{1,K}^2&\le C \left( h_K^{1+2s}\left| u\right| _{\frac{3}{2}+s,K}^2+(kh_K+1)^{2}k^4h_K^2\left\| u\right\| _{0,K}^2\right) , \end{aligned}$$

(59)

$$\begin{aligned} \left| \nabla (u-v_{p})\right| _{\frac{1}{2}+s,K}^{2}&\le C \left( \left| u\right| _{\frac{3}{2}+s,K}^{2} + (1+h_{K}k)^{4}h_{K}^{1-2s}k^{4}\left\| u\right\| _{0,K}^{2}\right) , \end{aligned}$$

(60)

with constants $C>0$ independent of $u$, $K$ and $L$ (depending only on $\rho _{0}$ and $\rho$ from Assumption (M1)).

Proof

Set $\hat{p}:=Q_{\hat{K}}^{2}\hat{u}$ and write $\hat{v}_{p}\in P\!W_{p,\hat{k}}(\hat{K})$, with $\hat{k}:=h_{K}k$, for the plane wave approximation of $\hat{p}$ according to Lemma 6.3. Its transformation back to $K$ provides $v_{p}\in P\!W_{p,k}(K)$. Simple transformations of norms yield

$$\begin{aligned} \left\| u-v_{p}\right\| _{0,K}= & {} h_{K}\left\| \hat{u}-\hat{v}_{p}\right\| _{0,\hat{K}} \le h_{K} \left( \left\| \hat{u}- \hat{p} \right\| _{0,\hat{K}} + \left\| \hat{p}-\hat{v}_{p}\right\| _{0,\hat{K}} \right) \\&\overset{(53)}{\le } C h_{K} \left( C\left| \hat{u}\right| _{\frac{3}{2}+s,\hat{K}} + \left\| \hat{p}-\hat{v}_{p}\right\| _{0,B_{1}} \right) \\&\overset{(55),(51)}{\le } C h_{K} \left( \left| \hat{u}\right| _{\frac{3}{2}+s,\hat{K}} + h_{K}^{2}k^{2}\left\| \hat{u}\right\| _{0,B_{\rho _{0}}}\right) \\\le & {} C \left( h_{K}^{\frac{3}{2}+s}\left| u\right| _{\frac{3}{2}+s,K} + h_{K}^{2}k^{2} \left\| u\right\| _{0,K}\right) . \end{aligned}$$

Rather similar arguments establish the second assertion of the lemma for the same $v_{p}$:

$$\begin{aligned} \left| u-v_{p}\right| _{1,K}= & {} \left| \hat{u}-\hat{v}_{p}\right| _{1,\hat{K}} \le \left| \hat{u}- \hat{p}\right| _{1,\hat{K}} + \left| \hat{p}-\hat{v}_{p}\right| _{1,\hat{K}} \\&\overset{(54)}{\le } C \left( \left| \nabla \hat{u}\right| _{\frac{1}{2}+s,\hat{K}} + \left| \hat{p}-\hat{v}_{p}\right| _{1,B_{1}} \right) \\&\overset{(57)}{\le } C \left( \left| \nabla \hat{u}\right| _{\frac{1}{2}+s,\hat{K}} + (h_{K}k+1)h_{K}^{2}k^{2} \left\| \hat{p}\right\| _{0,B_{1}}\right) \\&\overset{(51)}{\le } C \left( \left| \hat{u}\right| _{\frac{3}{2}+s,\hat{K}} + (h_{K}k+1)h_{K}^{2}k^{2} \left\| \hat{u}\right\| _{0,B_{\rho _{0}}} \right) \\\le & {} C \left( h_{K}^{\frac{1}{2}+s} \left| u\right| _{\frac{3}{2}+s,K} + (h_{K}k+1)h_{K} k^{2}\left\| u\right\| _{0,K} \right) . \end{aligned}$$

The third estimate follows along the same lines, using $\left| \nabla \hat{p}\right| _{\frac{1}{2}+s,\hat{K}}=|\hat{p}|_{2,\hat{K}}=0$:

$$\begin{aligned} \left| \nabla (u-v_{p})\right| _{\frac{1}{2}+s,K}= & {} h_{K}^{-\frac{1}{2}-s}\left| \nabla (\hat{u}-\hat{v}_{p})\right| _{\frac{1}{2}+s,\hat{K}} \\\le & {} h_{K}^{-\frac{1}{2}-s} \left( \left| \nabla (\hat{u}-\hat{p})\right| _{\frac{1}{2}+s,\hat{K}} + \left| \nabla (\hat{p}-\hat{v}_{p})\right| _{\frac{1}{2}+s,\hat{K}} \right) \\&\overset{(54)}{\le } C h_{K}^{-\frac{1}{2}-s} \left( \left| \nabla \hat{u}\right| _{\frac{1}{2}+s,\hat{K}} + \left\| \hat{p}-\hat{v}_{p}\right\| _{2,\hat{K}} \right) \\&\overset{(57)}{\le } C h_{K}^{-\frac{1}{2}-s} \left( \left| \nabla \hat{u}\right| _{\frac{1}{2}+s,\hat{K}} + (h_{K}k+1)^{2}h_{K}^{2}k^{2}\left\| \hat{p}\right\| _{0,B_{1}} \right) \\&\overset{(51)}{\le } C h_{K}^{-\frac{1}{2}-s} \left( \left| \hat{u}\right| _{\frac{3}{2}+s,\hat{K}} + (h_{K}k+1)^{2}h_{K}^{2}k^{2}\left\| \hat{u}\right\| _{0,B_{\rho _{0}}} \right) \\\le & {} C \left( \left| {u}\right| _{\frac{3}{2}+s, K} + (h_{K}k+1)^{2}h_{K}^{\frac{1}{2}-s}k^{2}\left\| u\right\| _{0,K} \right) . \end{aligned}$$

$\square$

The natural candidate for a local plane wave approximating $u$ on $K\in {\mathcal L}_{L}^{L}$ is $\left. v_{L}\right. _{|K} := v_{p}$ with $v_{p}$ supplied by the previous lemma. Then, we can tackle the terms on the right-hand side of (48) invoking Lemma 6.4 and the trace inequalities (19) and (20), respectively:

$$\begin{aligned} \frac{kh_{\max }}{h_K}\left\| u-v_p\right\| _{0,\partial K}^2&\overset{(19)}{\le } C \frac{kh_{\max }}{h_K}\left( \frac{1}{h_K}\left\| u-v_p\right\| _{0,K}^2+h_K\left| u-v_p\right| _{1,K}^2\right) \\\le & {} C {kh_{\max }}\left( \frac{1}{h_K^2}\left\| u-v_p\right\| _{0,K}^2+\left| u-v_p\right| _{1,K}^2\right) \\&\overset{(58),(59)}{\le } \!C {kh_{\max }}\left( h_K^{1+2s}\left| u\right| _{\frac{3}{2}\!+\!s,K}^2\!+\!(k^2h_K^2\!+\!1)k^4h_K^2\left\| u\right\| _{0,K}^2 \right) ,\\ \frac{1}{k}\left\| \nabla (u-v_p)\right\| _{0,\partial K}^2\le & {} \frac{C}{k} \left( h_{K}^{-1} \left\| \nabla (u-v_{p})\right\| _{0,K}^{2} + h_{K}^{2s}\left| \nabla (u-v_{p})\right| _{\frac{1}{2}+s,K}^{2}\right) \\&\overset{(59),(60)}{\le } \frac{C}{k} \left( h_{K}^{2s}\left| u\right| _{\frac{3}{2}+s,K}^{2} + (1+h_{K}k)^{4}h_{K}k^{4} \left\| u\right\| _{0,K}^{2}\right) . \end{aligned}$$

Therefore, taking into account the geometric scaling of the elements GM3, the contribution of $K$ to the right-hand side of (48) can be bounded as

$$\begin{aligned} \frac{1}{k}\big \Vert \nabla (u&-v_p)\cdot {{\mathbf {n}}}\Vert _{0,\partial K}^2+\frac{kh_{\max }}{h_K}\left\| u-v_p\right\| _{0,\partial K}^2\nonumber \\&\le C \sigma ^{2sL}\left( k^{-1}(1+k^2h_{\max }^2)\left| u\right| _{\frac{3}{2}+s,K}^{2} +h_{\max }^{1-2s}k^3(1+k^4h_{\max }^4) \left\| u\right\| _{0,K}^{2}\right) \nonumber \\&\le C \sigma ^{2sL} \, (1+k^4h_{\max }^4) \left( k^{-1}\left| u\right| _{\frac{3}{2}+s,K}^{2} + k^3 \left\| u\right\| _{0,K}^{2}\right) . \end{aligned}$$

(61)

6.5 Main a Priori Error Bound

Now we combine the estimates obtained in Sects. 6.3 and 6.4 into a final best approximation estimate for $u$ in $V_{L}$ in terms of the $DG^{+}$-norm, on families of geometric meshes complying with Assumptions (GM1)–(GM4). The focus is on asymptotic behaviour with respect to the depth $L$ of refinement. Hence, we do not look for the best possible $k$-dependence of the bounding constants. An explicit expression of the dependence on $k$, $\sigma$ and $u$ of the constant $\widetilde{C}$, the exponential rate $\widetilde{b}$ and the minimal number of layers $\widetilde{L}$ in the assertion of next theorem is shown in the proof.

Proof

Combining the result of Theorem 4.6 with (48), for all $\varepsilon >0$, we have with a constant $C>0$ independent of $L$, $k$, and $u$

$$\begin{aligned}&\left\| u-u_L\right\| _{0,\Omega }^2 \overset{\hbox {Thm. 4.6}}{\le } C\big (k^{-1}+k^{1+2\varepsilon }(C_{\mathcal F}+|\Gamma _R|)\big ) \inf \limits _{v_{L}\in V_{L}}\left\| u-v_{L}\right\| _{DG^{+}}^{2} \\&\quad \overset{(45),(48)}{\le }{C}\frac{k^{1+2\varepsilon }}{1-\sigma } \sum _{K\in {\mathcal T}_{L}} \inf \limits _{v_{L}\in P\!W_{p(L),k}(K)} \Big (k^{-1}\left\| \nabla (u-v_L)\cdot {{\mathbf {n}}}\right\| _{0,\partial K}^2 +\, \frac{kh_{\max }}{h_K}\left\| u-v_L\right\| _{0,\partial K}^2 \Big ) . \end{aligned}$$

Next, we split the sum into two parts comprising the small cells of layer ${\mathcal L}_{L}^{L}$ and cells away from corners, respectively:

$$\begin{aligned}&\sum _{K\in {\mathcal L}_{L}^{L}} \inf \limits _{v_{L}\in P\!W_{p(L),k}(K)} \Big ( k^{-1}\left\| \nabla (u-v_L)\cdot {{\mathbf {n}}}\right\| _{0,\partial K}^2+ \frac{kh_{\max }}{h_K}\left\| u-v_L\right\| _{0,\partial K}^2 \Big ) \\&\quad \overset{61}{\le } C \sigma ^{2sL} \, (1+k^4h_{\max }^4) \left( k^{-1}\left| u\right| _{\frac{3}{2}+s,K}^{2} + k^3 \left\| u\right\| _{0,K}^{2}\right) \\&\quad \overset{\hbox {Prop.~2.1}}{\le }C\sigma ^{2sL}\,k\,(1+k^4h_{\max }^4) \Big (\left\| g_{R}\right\| _{0,\Gamma _R}^2 +\left| g_{R}\right| _{s,\Gamma _{R}}^{2}\Big ), \end{aligned}$$

for all $s\in (0,\min \{s_\Omega ,r\})$, with $s_\Omega$ and $r$ as in Proposition 2.1. For an element $K$ away from corners, with $K_{\eta _{*}}$ as introduced in Lemma 6.2, $B(u) := (k\left\| u\right\| _{L^\infty (\Omega _{\eta _*})}+\left\| \nabla u\right\| _{L^\infty (\Omega _{\eta _*})})^{2}$, $\Omega _{\eta _{*}} := \bigcup \limits _{K\in {\mathcal T}_{L}\setminus {\mathcal L}_{L}^{L}}K_{\eta _{*}} \subset \mathcal {N}(u)$,

$$\begin{aligned}&\sum \limits _{K\in {\mathcal T}_{L}\setminus {\mathcal L}_{L}^{L}} \inf \limits _{v_{L}\in P\!W_{p(L),k}(K)} \Bigg (k^{-1}\left\| \nabla (u-v_L)\cdot {{\mathbf {n}}}\right\| _{0,\partial K}^2+ \frac{kh_{\max }}{h_K}\left\| u-v_L\right\| _{0,\partial K}^2 \Bigg )\\&\quad \overset{(49),(50)}{\le } C\,(kh_{\max })^{17}e^{6kh_{\max }}\,B(u) \sum \limits _{K\in {\mathcal T}_{L}\setminus {\mathcal L}_{L}^{L}} h_K \left[ e^{-2bL} + \frac{h_K^2\big (1+(kh_K)^{2L-2}\big )}{\big (c_0(L+1)\big )^{L}}\right] \\&\quad = C(kh_{\max })^{17}e^{6k h_{\max }}\,B(u) \left[ \sum \limits _{K\in {\mathcal T}_{L}\setminus {\mathcal L}_{L}^{L}}\! h_K e^{-2bL}\right. \\&\left. \qquad + \frac{1}{\big (c_0(L+1)\big )^{L}}\! \sum \limits _{K\in {\mathcal T}_{L}\setminus {\mathcal L}_{L}^{L}} \! h_K^3\big (1+(kh_K)^{2L-2}\big )\right] \\&\quad =: C\,(kh_{\max })^{17}e^{6k h_{\max }}\,B(u) \big [(I)+(II)\big ]. \end{aligned}$$

We bound separately $(I)$ and $(II)$:

$$\begin{aligned} (&I)=\sum _{\ell =0}^{L-1}\sum _{K\in {\mathcal L}^L_\ell } h_K\,e^{-2bL} \overset{(42)}{\le }C\left( \#{\mathcal L}_0^L+\sum _{\ell =1}^{L-1}\#{\mathcal L}_\ell ^L\,\sigma ^{\ell -1}(1-\sigma )\right) e^{-2bL}\\&\overset{(44)}{\le }C\left( 1+(1+\sigma )\sum _{\ell =0}^{L-2}\sigma ^{\ell }\right) e^{-2bL}\\&\le C\left( 1+\frac{(1+\sigma )(1-\sigma ^{L-1})}{(1-\sigma )}\right) e^{-2bL} \le \frac{C}{1-\sigma }\, e^{-2bL};\\ (&II)=\frac{1}{\big (c_0(L+1)\big )^{L}} \sum _{\ell =0}^{L-1}\sum _{K\in {\mathcal L}^L_\ell } h_K^3\big (1+(kh_K)^{2L-2}\big )\\&\overset{(42)}{\le }\frac{C \Big (\#{\mathcal L}_0^L(1+(kh_{\max })^{2L-2})+ \sum _{\ell =1}^{L-1}\#{\mathcal L}_\ell ^L\,\sigma ^{3\ell -3}(1-\sigma )^3 \big (1+(kh_{K})^{2L-2}\big )\Big )}{\big (c_0(L+1)\big )^{L}}\\&\overset{(44)}{\le }\frac{C \Big ((1+(kh_{\max })^{2L-2})+(1+\sigma )(1-\sigma )^2 \sum _{\ell =0}^{L-2}\sigma ^{3\ell }\big (1+(kh_{K})^{2L-2}\big )\Big ) }{\big (c_0(L+1)\big )^{L}}\\&\le \frac{C \left( (1+{(kh_{\max })}^{2L-2})+(1+\sigma )(1-\sigma )^2 \,\frac{1-\sigma ^{3L-3}}{1-\sigma ^3}\,\big (1+(kh_{\max }) ^{2L-2}\big ) \right) }{\big (c_0(L+1)\big )^{L}}\\&\overset{{\frac{1-\sigma }{1-\sigma ^m}\le 1}}{\le } \frac{C}{\big (c_0(L+1)\big )^{L}}\, \big (1+(k h_{\max })^{2L-2}\big ) \le C (c_0\,L)^{-L}\big (1+(k h_{\max })^{2L}\big )\\&= C\,\big ((c_0^2\,L)^{-L/2}+ (c_0^2\,L/(k h_{\max })^4)^{-L/2}\big )\, L^{-L/2}\\&\le C\,\big (e^{1/(2c_0^2 e)}+e^{(k h_{\max })^4/(2c_0^2 e)}\big ) L^{-L/2}, \end{aligned}$$

where in the last step, we have used the bound $(aL)^{-L/2}\le e^{1/(2ea)}$ which holds for all $a,L>0$.

Combining the above estimates, taking into account that $B(u)\le C\,k^{10} e^{k/2e}$ due to Lemma 2.4, gives

$$\begin{aligned} \left\| u-u_L\right\| _{0,\Omega }^2 \le&\, C\,\frac{k^{1+2\varepsilon }}{1-\sigma } \big (1+(kh_{\max })^{17}\big ) \Bigg (k\,\sigma ^{2sL} +k^{10}\,e^{6kh_{\max }+k/2e}\bigg (\frac{e^{-2bL}}{1-\sigma }\\&+ \big (e^{1/(2c_0^2 e)}+e^{(k h_{\max })^4/(2c_0^2 e)}\big ) L^{-L/2}\bigg )\Bigg ), \end{aligned}$$

where we have incorporated in $C$ the dependence on $g_R$. We have

$$\begin{aligned} \sigma ^{2sL}=e^{-L(-2s\log \sigma )},\qquad L^{-L/2}=e^{-L(\log \sqrt{L})}, \end{aligned}$$

Assuming $\widetilde{L}\ge e^{4b}$, all the exponentials are bounded from above by $e^{-2\min \{-s\log \sigma ,b\}L}$. Thus,

$$\begin{aligned} \left\| u-u_L\right\| _{0,\Omega }\le C(k)\,e^{-\min \{-s\log \sigma ,b\} L}, \end{aligned}$$

with

$$\begin{aligned} C(k)\le \frac{C^\prime \,\big [k^{6} e^{k/4e}\big ] \Big [\big (1+(kh_{\max })^{17/2}\big ) e^{3kh_{\max }+{\max \{1,(k h_{\max })^4\}}/(4c_0^2e)}\Big ]}{1-\sigma }, \end{aligned}$$

with $C^\prime$ independent of $\sigma$, $k$ and $L$. Since, by (46) and (47), $L\ge C((1-\sigma )\dim V_L)^{\frac{1}{2}}$, with $C$ only depending on the constants appearing in Assumptions (M1), (GM2) and (GM3), the assertion of the theorem follows with

$$\begin{aligned} \widetilde{b}=C\sqrt{1-\sigma }\min \{-s\log \sigma ,b\}. \end{aligned}$$

$\square$

The proof of Theorem 6.5 shows that the rate $\widetilde{b}$ of exponential convergence of the Trefftz-DG method and the layer number threshold $\widetilde{L}$ depend only on: (i) the regularity parameter $s$ relative to the solution $u$; (ii) the mesh grading parameter $\sigma$; (iii) the parameter $b$ from Proposition 5.1 (and [20], Corollary 4.11]), which is the exponential convergence rate for the approximation of certain harmonic functions by harmonic polynomials and which in turn depends on the star-shapedness parameters $\rho$ and $\rho _0$ in Assumption (M1) and again on $\sigma$ via Lemma 6.2.

Remark 6.6

If we monitor the dependence on $k$ throughout the proof of Theorem 6.5, we see that if the “scale resolution” condition $kh_{\max }\le 1$ on the initial mesh ${\mathcal T}_1$ is satisfied, then the constant $\widetilde{C}$ in the error bound of the theorem grows in $k$ as $k^6e^{k/4e}$. The bound in Lemma 2.4 on the analytic extension of $u$ is responsible for a factor $k^5e^{k/4e}$; we expect that a refinement of this argument might make the constant of the final bound of Theorem 6.5 linear in $k$ under the above scale resolution condition.

If the scale resolution condition is not satisfied, the constant $\widetilde{C}$ may increase like $\exp (k^{4})$ for $k\rightarrow \infty$. (We note that the more than exponential term in $k$ only appears multiplied to the fastest converging term in $L$, i.e. $L^{-L/2}$.) This bound can easily be improved to $\exp (k^{2+\epsilon })$ for any $\epsilon >0$ (substituting the assumption $\log L\ge 4b$ with $\log L\ge 2(2+\epsilon )b/\epsilon$). However, we believe that also this prediction is way too pessimistic.

Remark 6.7

The Trefftz-DG method with a basis composed by circular waves (i.e. Fourier–Bessel functions) can be considered in the same setting examined here (graded meshes, flux parameters). Using Proposition 5.3 instead of Proposition 5.4, the same exponential convergence in the square root of the total number of degrees of freedom, as in the plane wave basis case, is achieved.

Remark 6.8

For piecewise polynomial $hp$-approximation, it is possible to use local degrees on $K\in {\mathcal L}_{\ell }^{L}$ linearly increasing with $L-\ell$ without affecting overall exponential convergence [15, 37]. If $\sigma$ is sufficiently small, the same result is possible in the present setting, by a slight modification of the analysis of Sect. 6. If in (46) we chose to use

$$\begin{aligned} p(L,\ell ):=2(L-\ell )+5 \end{aligned}$$

plane waves in each element $K\in {\mathcal L}_\ell ^L$ (recall that we need $p\ge 5$ in Lemma 6.4), the bounds of the terms $(I)$ and $(II)$ in the proof of Theorem 6.5, which are the crucial points to obtain the exponential convergence in $L$, are modified as follows, provided that $\sigma <e^{-2b}$,

$$\begin{aligned} (I)&=\!\sum _{\ell =0}^{L-1}\sum _{K\in {\mathcal L}^L_\ell } h_K\,e^{-2b(L-\ell +2)} \overset{(42)}{\le }\! C\Big (\#{\mathcal L}_0^L\!+\!\sum _{\ell =1}^{L-1}\#{\mathcal L}_\ell ^L\,\sigma ^{\ell -1}(1-\sigma )e^{2b\ell }\Big ) e^{-2b(L+2)}\\&\overset{(44)}{\le }C\Big (1+(1+\sigma )\sum _{\ell =0}^{L-2}(\sigma e^{2b})^{\ell }\Big ) e^{-2b(L+2)} \le \frac{C}{1-\sigma e^{2b}}\, e^{-2b(L+2)},\\ (II)&=\sum _{K\in {\mathcal T}_L\setminus {\mathcal L}_L^L}\frac{h_K^3\big (1+(kh_K)^{2(L-\ell )+2}\big )}{\big (c_0(L-\ell +3)\big )^{L-\ell +2}}\\&\le \frac{C(1+(kh_{\max })^{2L+2})}{\big (c_0(L+3)\big )^{L+2}} +C \sum _{\ell =1}^{L-1}\frac{e^{-6b(\ell -1)}\big (1+(kh_K)^{2(L-\ell )+2}\big )}{\big (c_0(L-\ell +3)\big )^{L-\ell +2}}\\ \end{aligned}$$

The first term in $(II)$ can be bounded with an exponential by proceeding as in the proof of Theorem 6.5. For the second term, simply using the partial sum of the exponential function and $(m+2)^{-m}\le 1/m!$, we have

$$\begin{aligned}&\sum _{\ell =1}^{L-1}\frac{e^{-6b(\ell -1)}\big (1+(kh_K)^{2(L-\ell )+2}\big )}{\big (c_0(L-\ell +3)\big )^{L-\ell +2}}\\&\quad \overset{(m:=L-\ell +1)}{=} \sum _{m=2}^{L}\frac{e^{-6b(L-m)}\big (1+(kh_K)^{2m}\big )}{\big (c_0(m+2)\big )^{m+1}}\\&\quad \le \frac{e^{-6bL}}{2c_0}\sum _{m=2}^{L}\left( \frac{e^{6b}\,\max \{1,(kh_{\max })^2\}}{c_0}\right) ^m \left( \frac{1}{m+2}\right) ^m \\&\quad \le \left( \frac{e^{-6bL}}{2c_0} \,e^{e^{6b}\max \{1,(kh_{\max })^2\}/c_0}\right) \,e^{-6bL}. \end{aligned}$$

Such a reduction of the plane wave number in the small elements near the corners seems to be inevitable in practice to curb the instability of the plane wave basis, see [22].