Non-conforming Harmonic Virtual Element Method: $h$- and $p$-Versions

Abstract

We study the $h$- and $p$-versions of non-conforming harmonic virtual element methods (VEM) for the approximation of the Dirichlet–Laplace problem on a 2D polygonal domain, providing quasi-optimal error bounds. Harmonic VEM do not make use of internal degrees of freedom. This leads to a faster convergence, in terms of the number of degrees of freedom, as compared to standard VEM. Importantly, the technical tools used in our $p$-analysis can be employed as well in the analysis of more general non-conforming finite element methods and VEM. The theoretical results are validated in a series of numerical experiments. The hp-version of the method is numerically tested, demonstrating exponential convergence with rate given by the square root of the number of degrees of freedom.

1 Introduction

In recent years, Galerkin methods based on polygonal/polyhedral meshes have attracted a lot of attention, owing to their flexibility in dealing with complex geometries [5, 19, 31, 38, 39, 41, 55]. In this paper, we focus on the virtual element method (VEM) introduced in [13, 16]. The main feature of VEM, in addition to the fact that they allow for general polytopal meshes, is that they are based on trial and test spaces that consist of solutions to local problems mimicking the target one. These functions are not known in a closed form, which is at the origin of the name “virtual”. Importantly, the construction of the method does not rely on an explicit representation of the basis functions, but rather on the explicit knowledge of the degrees of freedom. This allows to compute certain projection operators from local VE spaces into polynomial ones, which are instrumental in the definition of proper bilinear forms.

Owing to its flexibility and simplicitity of the implementation, despite its novelty, the basic VEM paradigm has already been extended to highly-regular [22] and non-conforming [7, 34] approximating spaces, combined with domain decomposition techniques [27], adaptive mesh refinement [32], adapted to curved domains [23], and applied to a wide variety of problems; among them, we recall general second-order elliptic problems [14], eigenvalue problems [43, 53], Stokes problem [21], elasticity problem [15], Helmholtz problem [54], Cahn–Hilliard equation [3], discrete fracture network simulations [24], and topology optimization [42].

In this paper, we introduce and analyze non-conforming harmonic VEM for the approximation of the Dirichlet–Laplace problem on polygonal domains. These methods can be seen as the intermediate conformity level between the continuous harmonic VEM developed in [36], and the harmonic discontinuous Galerkin finite element method (DG-FEM) of [45,46,47]. As typically done in non-conforming methods, instead of requiring $C^0$-continuity of test and trial functions over the entire physical domain, one only imposes that the moments, up to a certain order, of their jumps across two adjacent elements are zero. We highlight that non-conforming VEM were introduced in [7] for the approximation of the Poisson problem and were subsequently extended to the approximation of general elliptic and Stokes problems in [33, 34], respectively. Our method inherits the structure of that of [7], but makes use of harmonic basis functions, which yield to faster convergence, when approximating harmonic solutions, as compared to standard basis functions.

We are particularly interested in the investigation of the $h$- and $p$-versions of these methods. In the former version, convergence is achieved by fixing the dimension of local spaces and refining the mesh, whereas, in the latter, by fixing a single mesh and increasing the dimension of local spaces. A combination of the two goes by the name of $hp$-version. The literature regarding the $p$- and $hp$-versions of VEM is restricted to [4, 17, 18, 40, 48], in addition to the above-mentioned work [36]; for the $hp$-version of DG-FEM and hybrid-high order methods on polytopal grids, see [1, 31] and the references therein. We derive quasi-optimal error bounds in the broken $H^1$ norm and in the $L^2$ norm, which are explicit in terms of the mesh size and of the degree of accuracy of the method. Although not covered by our theoretical analysis, we provide numerical evidence that, similarly as for the harmonic VEM and harmonic DG-FEM [36, 45], the exponential convergence of the $hp$-version of the non-conforming harmonic VEM is faster than the one of standard FEM [8, 56] and VEM [17, 18].

The tools that we employ in the forthcoming $p$-analysis of non-conforming harmonic VEM can actually be employed as well in the $p$-analysis of non-conforming FEM and of non-conforming VEM. For instance, our argument to trace back best approximation estimates by means of non-conforming harmonic VE functions to best approximation estimates by means of discontinuous harmonic polynomials (Proposition 3.1) extends to the non-harmonic case (Proposition 3.8). This provides a useful tool in order to develop a p-analysis of the non-conforming VEM of [7].

We stress that, in the high-order case, the construction of an explicit basis for non-conforming harmonic VEM, as well as for non-conforming standard VEM, is much simpler than for non-conforming FEM, see for instance [37].

The design and analysis of the non-conforming harmonic VEM developed in this paper pave the way for the study of VEM for the Helmholtz problem in a truly Trefftz setting, alternative to the conforming plane wave VEM of [54], which was based on a partition of unity approach. In fact, the non-conforming framework seems to be the most appropriate one in order to design virtual Helmholtz–Trefftz approximation spaces. Such an extension has been investigated in the recent work [49].

The outline of this paper is as follows. In Sect. 2, the model problem is formulated and the concept of regular polygonal decompositions needed for the analysis is introduced; besides, we recall the definition of non-conforming Sobolev spaces subordinated to polygonal decompositions of the physical domain. Section 3 is dedicated to the construction of the 2D non-conforming harmonic VEM and to the analysis of its $h$- and $p$-versions; further, a hint for the extension to the 3D case is given. Next, in Sect. 4, numerical results validating the theoretical convergence estimates are presented; a numerical investigation of the full $hp$-version of the method is also provided. Finally, details on the implementation of the method are given in “Appendix A”.

Notation We fix here once and for all the notation employed throughout the paper. Given any domain $D \subseteq \mathbb {R}^d$, $d \in \mathbb N$, and $\ell \in \mathbb N$, we denote by $\mathbb P_\ell (D)$ and $\mathbb H_\ell (D)$ the spaces of polynomials and harmonic polynomials up to order $\ell$ over D, respectively; moreover, we set $\mathbb P_{-1}(D) = \mathbb H_{-1}(D) = \emptyset$.

We use the standard notation for Sobolev spaces, norms, seminorms and inner products. More precisely, we denote the Sobolev space of functions with square integrable weak derivatives up to order s over D by $H^s(D)$, and the corresponding seminorms and norms by $|\cdot |_{s,D}$ and $||\cdot ||_{s,D}$, respectively. Sobolev spaces of non-integer order can be defined, for instance, by interpolation theory. In addition, for bounded D, $H^{1/2}(\partial D)$ denotes the space of traces of $H^1(D)$ functions; $H^1_0(D)$ and $H^1_g(D)$ are the Sobolev spaces of $H^1$ functions with traces equal to zero and equal to a given function $g\in H^{1/2}(\partial D)$, respectively. Further, $(\cdot ,\cdot )_{0,D}$ is the usual $L^2$ inner product over D.

We employ the following multi-index notation: for $\varvec{\alpha }=(\alpha _1,\dots ,\alpha _d)$,

$$\begin{aligned} \varvec{x}^{\varvec{\alpha }} = x_1^{\alpha _1} x_2^{\alpha _2} \dots x_d^{\alpha _d}, \quad \partial ^{\varvec{\alpha }} = \partial _1^{\alpha _1} \partial _2^{\alpha _2} \dots \partial _d^{\alpha _d}, \end{aligned}$$

with $|\varvec{\alpha }|=\alpha _1+\dots +\alpha _d$, and where $\partial _\ell ^\alpha$ denotes the $\alpha$-th partial derivative along direction $x_\ell$.

In the sequel, we also use the notation $a \lesssim b$ meaning that there exists a constant $c>0$, independent of $h$ and $p$, such that $a \leqslant c\, b$. Finally, we use the notation $a \approx b$ in lieu of $a \lesssim b$ and $b \lesssim a$ simultaneously.

2 Continuous Problem, Polygonal Decompositions and Functional Setting

Here, we want to set the target problem and some basic notation we need for the construction of the non-conforming harmonic virtual element method (VEM). More precisely, the outline of the section is as follows. In Sect. 2.1, we introduce the model problem, that is a Laplace problem on a polygonal domain. Then, in Sect. 2.2, we define the concept of regular decompositions into polygons of the physical domain of the problem. Finally, in Sect. 2.3, we describe non-conforming Sobolev spaces over such polygonal decompositions.

2.1 The Continuous Problem

The target problem we aim to approximate is a Laplace problem over a polygonal domain $\Omega \subset \mathbb R^2$ with boundary $\partial \Omega$. More precisely, given $g\in H^{1/2}(\partial \Omega )$, we look for a function u solving

(1)

The weak formulation of (1) reads

$$\begin{aligned} {\left\{ \begin{array}{ll} \text {find } u\in V_g\text { such that}\\ a(u,v) = 0\quad \forall v\in V_0,\\ \end{array}\right. } \end{aligned}$$

(2)

where

$$\begin{aligned} a(u,v) := (\nabla u, \nabla v)_{0,\Omega },\quad V_g:= H^1_g(\Omega ),\quad V_0:= H^1_0(\Omega ). \end{aligned}$$

(3)

Well-posedness of problem (2) follows from a lifting argument and the Lax–Milgram lemma.

2.2 Regular Polygonal Decompositions

In this section, we introduce the concept of regular sequences of polygonal decompositions of the domain $\Omega$, which will be needed in the forthcoming analysis of the method.

Let $\{\mathcal T_n\}_{n\in \mathbb N}$ be a sequence of conforming polygonal decompositions of $\Omega$; by conforming, we mean that, for each $n\in \mathbb N$, every internal edge $e$ of $\mathcal T_n$ is contained in the boundary of precisely two elements in the decomposition. This automatically includes the possibility of dealing with hanging nodes.

For all $n\in \mathbb N$, with each $\mathcal T_n$, we associate $\mathcal E_n$, $\mathcal E_n^I$ and $\mathcal E_n^B$, which denote its set of edges, internal edges and boundary edges, respectively. Moreover, with each element $K$ of $\mathcal T_n$, we associate $\mathcal E^K$, the set of its edges. Finally, we set for all $K\in \mathcal T_n$ and for all $n \in \mathbb N$,

$$\begin{aligned} h_K:= \text {diam}(K), \; \quad h:= \max _{K\in \mathcal T_n} h_K, \quad h_e:= \text {length}(e), \, \forall e\in \mathcal E^K, \end{aligned}$$

and we denote by $\mathbf x_K$ the centroid of $K$.

We say that $\{\mathcal T_n\}_{n\in \mathbb N}$ is a regular sequence of polygonal decompositions if the following assumptions are satisfied:

(D1):

there exists a positive constant $\rho _1$ such that, for all $n\in \mathbb N$ and for all $K\in \mathcal T_n$, $h_e\ge \rho _1 h_K$ for all edges $e$ of $K$;

(D2):

there exists a positive constant $\rho _2$ such that, for all $n\in \mathbb N$ and for all $K\in \mathcal T_n$, $K$ is star-shaped with respect to a ball of radius greater than or equal to $\rho _2 h_K$.

The assumptions (D1) and (D2) imply the following property:

(D3):

there exists a constant $\Lambda \in \mathbb N$ such that, for all $n\in \mathbb N$ and for all $K\in \mathcal T_n$, card($\mathcal E^K$)$\le \Lambda$, that is, the number of edges of each element is uniformly bounded.

We point out that, in this definition, we are not requiring any quasi-uniformity on the size of the elements. A discussion of VEM under more general mesh assumptions is the topic of [20, 29].

Remark 1

In the forthcoming analysis, we will employ a number of standard functional inequalities (such as the Poincaré inequality and trace inequalities). It can be proven that the constants appearing in such inequalities depend solely on the parameters $\rho _1$, $\rho _2$, and $\Lambda$ introduced in (D1)–(D3). We will omit such a dependence, for ease of notation.

For future use, we also define local bilinear forms on polygons $K\in \mathcal T_n$ as

$$\begin{aligned} a^K(u,v) := (\nabla u, \nabla v)_{0,K} \quad \forall u,\,v\in H^1(K). \end{aligned}$$

(4)

2.3 Non-conforming Sobolev Spaces

Having introduced the concept of regular sequences of meshes, we pinpoint the concept of sequences of broken and non-conforming Sobolev spaces, along with their norms. For all $n\in \mathbb N$ and $s>0$, we define the broken Sobolev spaces on $\mathcal T_n$ as

$$\begin{aligned} H^{s}(\mathcal T_n) := \left\{ v\in L^2(\Omega ) \mid v_{|_K} \in H^s(K) \ \forall K\in \mathcal T_n\right\} \end{aligned}$$

and the corresponding broken seminorms and norms

$$\begin{aligned} \vert v\vert ^2_{s,\mathcal T_n} := \sum _{K\in \mathcal T_n} \vert v\vert _{s,K}^2, \quad \Vert v\Vert ^2_{s,\mathcal T_n} := \sum _{K\in \mathcal T_n} \Vert v\Vert _{s,K}^2. \end{aligned}$$

(5)

Particular emphasis is stressed on the broken $H^1$ bilinear form

$$\begin{aligned} (u,v) _{1,\mathcal T_n} := \sum _{K\in \mathcal T_n} (\nabla u,\nabla v)_{0,K}. \end{aligned}$$

In order to define non-conforming Sobolev spaces associated with polygonal decompositions, we need to fix some additional notation. In particular, given any internal edge $e\in \mathcal E_n^I$ shared by the polygons $K^-$ and $K^+$ in $\mathcal T_n$, we denote by $\mathbf n_{K^\pm }^e$ the two outer normal unit vectors with respect to $K^{\pm }$. For simplicity, we will later only write $\mathbf n_{K^\pm }$ instead of $\mathbf n_{K^\pm }^e$. Moreover, for boundary edges $e\in \mathcal E_n^B$, we introduce the normal unit vector $\mathbf n_\Omega$ pointing outside $\Omega$. Having this, for any $v\in H^1(\mathcal T_n)$, we set the jump operator across an edge $e\in \mathcal E_n$ to

$$\begin{aligned} \llbracket v\rrbracket := {\left\{ \begin{array}{ll} v_{|_{K^+}} \mathbf n_{K^+} + v_{|_{K^-}} \mathbf n_{K^-}&{}\quad \text {if } e\in \mathcal E^I_n\\ v\mathbf n_\Omega &{} \quad \text {if } e\in \mathcal E^B_n.\\ \end{array}\right. } \end{aligned}$$

(6)

Finally, we introduce the global non-conforming Sobolev space of order $k\in \mathbb N$ with respect to the decomposition $\mathcal T_n$ incorporating boundary conditions in a non-conforming sense: Given $g\in H^{1/2}(\partial \Omega )$ and $k\in \mathbb N$, we define

$$\begin{aligned} \begin{aligned} H_{g}^{1,nc} (\mathcal T_n, k) := \{ v\in H^1(\mathcal T_n) \, \mid \,&\int _e\llbracket v\rrbracket \cdot \mathbf n\, q_{k-1} \, d s= 0 \quad \forall q_{k-1} \in \mathbb {P}_{k-1}(e), \ \forall e\in \mathcal E_n^I,\\&\int _e\llbracket v\rrbracket \cdot \mathbf n\, q_{k-1} \, d s= \int _egq_{k-1} \, d s\quad \forall q_{k-1} \in \mathbb {P}_{k-1}(e), \ \forall e\in \mathcal E_n^B \},\\ \end{aligned} \end{aligned}$$

(7)

where $\mathbf n$ is either of the two normal unit vectors to e, but fixed, if $e \in \mathcal {E}_n^I$, and $\mathbf n=\mathbf n_\Omega$, if $e \in \mathcal {E}_n^B$. In the homogeneous case, definition (7) becomes

$$\begin{aligned} \begin{aligned} H^{1,nc}_0(\mathcal T_n, k) := \left\{ v\in H^1(\mathcal T_n) \, \mid \, \int _e\llbracket v\rrbracket \cdot \mathbf n\, q_{k-1} \, d s= 0 \quad \forall q_{k-1} \in \mathbb {P}_{k-1}(e), \ \forall e\in \mathcal E_n \right\} .\\ \end{aligned} \end{aligned}$$

(8)

Importantly, the seminorm $\vert \cdot \vert _{1,\mathcal T_n}$ is actually a norm for functions in $H^{1,nc}_0 (\mathcal T_n, k)$. In [28], the validity of the following Poincaré  inequality was proven: there exists a positive constant $c_P$ only depending on $\Omega$ such that, for all $k\in \mathbb N$,

$$\begin{aligned} \Vert v\Vert _{0,\Omega } \le c_P \vert v\vert _{1,\mathcal T_n} \quad \forall v\in H^{1,nc}_0(\mathcal T_n, k). \end{aligned}$$

(9)

3 Non-conforming Harmonic Virtual Element Methods

In this section, we introduce a non-conforming harmonic virtual element method for the approximation of problem (2) and investigate its $h$- and $p$-versions. To this purpose, in addition to (D1)–(D3), we will also require on the sequence of meshes $\{\mathcal T_n\}_{n\in \mathbb N}$ the following quasi-uniformity assumption:

(D4):

there exists a constant $\rho _3 \ge 1$ such that, for all $n \in \mathbb N$ and for all $K_1$ and $K_2$ in $\mathcal T_n$, it holds $h_{K_2} \le \rho _3 h_{K_1}$.

We want to approximate problem (2) with a method of the following type:

$$\begin{aligned} {\left\{ \begin{array}{ll} \text {find } u_n\in V^{\Delta ,p}_{n,g}\text { such that}\\ a_n(u_n, v_n) = 0 \quad \forall v_n\in V^{\Delta ,p}_{n,0},\\ \end{array}\right. } \end{aligned}$$

(10)

where the space of trial functions $V^{\Delta ,p}_{n,g}$ and the space of test functions $V^{\Delta ,p}_{n,0}$ are finite dimensional (non-conforming) spaces on a mesh $\mathcal T_n$, “mimicking” the infinite dimensional spaces $V_g$ and $V_0$, defined in (3), respectively. Moreover, $a_n(\cdot ,\cdot ): V^{\Delta ,p}_{n,g}\times V^{\Delta ,p}_{n,0}\rightarrow \mathbb R$ is a computable discrete bilinear form mimicking its continuous counterpart defined again in (3). Such approximation spaces and discrete bilinear forms have to be tailored so that method (10) is well-posed and provides “good” $h$- and $p$-approximation estimates.

The outline of this section is as follows. We first introduce suitable global approximation spaces $V^{\Delta ,p}_{n,g}$ and $V^{\Delta ,p}_{n,0}$ in Sect. 3.1, highlighting their approximation properties in Sect. 3.2. Next, in Sect. 3.3, we define and provide an explicit discrete bilinear form and, moreover, we discuss its properties. An abstract error analysis is carried out in Sect. 3.4; such analysis is instrumental for the $h$- and $p$-error estimates proved in Sect. 3.5. $L^2$ error bounds are provided in Sect. 3.6. Finally, in Sect. 3.7, we give a hint concerning the extension to the 3D case and we stress the main differences between the 2D and 3D cases. Some details on the implementation of the method are presented in “Appendix A”.

3.1 Non-conforming Harmonic Virtual Element Spaces

The aim of the present section is to introduce non-conforming harmonic virtual element spaces with uniform degree of accuracy. To this purpose, we begin with the description of the local harmonic VE spaces, modifying those in [36] into a new setting suited for building global non-conforming spaces.

Let $p\in \mathbb N$ be a given parameter. For all $n\in \mathbb N$ and for all $K\in \mathcal T_n$, we set

$$\begin{aligned} V^\Delta (K):= \left\{ v_n\in H^1(K) \mid \Delta v_n= 0 \text { in } K,\, \left( \nabla v_n\cdot \mathbf n_K\right) _{|_e} \in \mathbb P_{p-1}(e) \ \forall e\in \mathcal E^K \right\} . \end{aligned}$$

(11)

In words, $V^\Delta (K)$ consists of harmonic functions with piecewise (discontinuous) polynomial normal traces on the boundary of $K$.

The space $V^\Delta (K)$ has dimension $N_Kp$, $N_K$ being the number of edges of $K$. A set of $N_Kp$ degrees of freedom for $V^\Delta (K)$ is the following. Given $v_n\in V^\Delta (K)$,

$$\begin{aligned} \frac{1}{h_e} \int _ev_nm_{r}^e\, d s\quad \forall r=0,\dots , p-1, \, \forall e\in \mathcal E^K , \end{aligned}$$

(12)

where $\{m_r^e\}_{r=0,\ldots ,p-1}$ is any basis of $\mathbb P_{p-1}(e)$. These degrees of freedom are in fact unisolvent since, if $v_n\in V^\Delta (K)$ has all the degrees of freedom equal to 0, then

$$\begin{aligned} |\nabla v_n|_{1,K}^2 = \int _K(\underbrace{-\Delta v_n}_{=0}) \, v_n\, d x+ \int _{\partial K} \left( \nabla v_n\cdot \mathbf n_K\right) \, v_n\, d s= \sum _{e\in \mathcal E^K} \int _e\underbrace{ \left( \nabla v_n\cdot \mathbf n_K\right) }_{\in \mathbb P_{p-1}(e)} v_n\, d s= 0, \end{aligned}$$

which implies that $v_n$ is constant. This, in addition to

$$\begin{aligned} h_e v_n= \int _ev_n\, d s= \int _e1\, v_n\, d s=0 , \end{aligned}$$

for some edge $e\in \mathcal E^K$, implies $v_n=0$, providing unisolvence.

We denote by $\{\varphi _{j,r}\}_{\begin{array}{c} j=1\ldots ,N_K\\ r=0,\ldots , p-1 \end{array}}$ the local canonical basis associated with the set of degrees of freedom (12), namely

$$\begin{aligned} \text {dof}_{i,s}\left( \varphi _{j,r}\right) = {\left\{ \begin{array}{ll} 1 &{} \quad \text {if } i=j \ \text {and} \ s=r\\ 0 &{} \quad \text {otherwise}\\ \end{array}\right. }\quad \forall \, i,j =1,\dots ,N_K,\, \forall \, s,r=0,\dots ,p-1. \end{aligned}$$

(13)

We underline that the indices i and j refer to the edge, whereas the indices s and r refer to the polynomial $m_{r}^e$ employed in the definition of the local degrees of freedom (12).

It is worth to note that the local canonical basis consists of functions that are not explicitly known inside the element and even their polynomial normal traces over the boundary are unknown.

By employing the degrees of freedom defined in (12), it is possible to compute the following two projectors. The first one is the edge $L^2$ projector onto the space of polynomials of degree $p-1$

$$\begin{aligned} \begin{aligned}&\Pi ^{0,e}_{p-1}: V^\Delta (K)|_e\rightarrow \mathbb P_{p-1}(e), \\&\quad \int _e\left( v_n- \Pi ^{0,e}_{p-1}v_n\right) q^e_{p-1}\ d s= 0\quad \forall v_n\in V^\Delta (K), \, \forall q^e_{p-1}\in \mathbb P_{p-1}(e). \end{aligned} \end{aligned}$$

(14)

The second one is the bulk $H^1$ projector onto the space of harmonic polynomials of degree $p$

$$\begin{aligned} \begin{aligned}&\Pi ^{\nabla ,\Delta ,K}_p= \Pi ^{\nabla ,K}_p: V^\Delta (K)\rightarrow \mathbb H_p(K),\\&\quad \int _K\nabla \left( v_n- \Pi ^{\nabla ,K}_pv_n\right) \cdot \nabla q^\Delta _p\, d x= 0\quad \forall v_n\in V^\Delta (K),\, \forall q^\Delta _p\in \mathbb H_p(K), \\&\quad \int _{\partial K} \left( v_n- \Pi ^{\nabla ,K}_pv_n\right) \, d s= 0\quad \forall v_n\in V^\Delta (K), \end{aligned} \end{aligned}$$

(15)

where the last condition is imposed in order to define the projector in a unique way.

We are ready to define global non-conforming harmonic VE spaces, which incorporate Dirichlet boundary conditions in a “non-conforming sense”. Let $p\in \mathbb N$ be a given parameter. Then, for any $g\in H^{1/2}(\partial \Omega )$, we set

$$\begin{aligned} V^{\Delta ,p}_{n,g}:= \left\{ v_n\in H^{1,nc}_{g} (\mathcal T_n,p) \, \mid \, v_{n|_K} \in V^\Delta (K)\ \forall K \in \mathcal T_n\right\} . \end{aligned}$$

(16)

We observe the following facts:

• Definition (16) includes the space of test functions $V^{\Delta ,p}_{n,0}$, by selecting $g=0$.

• The parameter p in (16) indicates the level of non-conformity of the method. The fact that the non-conformity is defined with respect to Dirichlet traces allows us to easily couple the local degrees of freedom into a global set, provided that we choose the same value p for the non-conformity parameter and for the polynomial degree entering definition (11) of the local spaces. The resulting global set of degrees of freedom is of dimension $\text {card}(\mathcal E_n)p$.

• Dirichlet boundary conditions on $\partial \Omega$ are imposed weakly via the definition of the non-conforming spaces (7) and (8). For instance, given a Dirichlet datum $g$, on all boundary edges $e\in \mathcal E_n^B$, we set

  $$\begin{aligned} \int _e\llbracket v_n\rrbracket \cdot \mathbf n_\Omega \, q^e_{p-1}\, d s= \int _ev_nq^e_{p-1}\, d s= \int _egq^e_{p-1}\, d s\quad \forall v_n\in V^{\Delta ,p}_{n,g},\ \forall q^e_{p-1}\in \mathbb P_{p-1}(e). \end{aligned}$$

Remark 2

We highlight that, at the discrete level, one should also take into account the approximation of the Dirichlet boundary condition $g$. In practice, assuming $g\in H^{\frac{1}{2} + \varepsilon }(\partial \Omega )$, for any $\varepsilon >0$ arbitrarily small, and denoting by $g_p$ the approximation of $g$ obtained by interpolating $g$ at the $p+1$ Gauß–Lobatto nodes on each edge in $\mathcal E_n^B$, one should define the trial space as

$$\begin{aligned} V^{\Delta ,p}_{n,g}:= \left\{ v_n\in H^{1,nc}_{g_p} (\mathcal T_n,p) \, \mid \, v_{n_{|_K}} \in V^\Delta (K)\ \forall K \in \mathcal T_n\right\} . \end{aligned}$$

With this definition, in the forthcoming analysis (see Propositions 3.1, 3.8, Theorems 3.3, 3.6, and 3.9 below), an additional term related to the approximation of the Dirichlet datum via Gauß–Lobatto interpolants should be taken into account. However, following [26, Theorems 4.2, 4.5], it is possible to show that the $h$- and $p$-rates of convergence of the method are not spoilt by this term. For this reason and for the sake of simplicity, we will neglect in the following the presence of this term and assume that the approximation space is the one defined in (16).

3.2 Approximation Properties of Functions in Non-conforming Harmonic Virtual Element Spaces

In this section, we deal with approximation properties of functions in the non-conforming harmonic VE spaces $V^{\Delta ,p}_{n,g}$ and $V^{\Delta ,p}_{n,0}$.

Since $h$- and $p$-approximation properties of harmonic functions via harmonic polynomials are known, see e.g. [11, 45], we want to relate best approximation estimates in the non-conforming harmonic VE spaces to the corresponding ones in discontinuous harmonic polynomial spaces. In particular, we prove the following result.

Proposition 3.1

Given $g\in H^{1/2}(\partial \Omega )$, let $u\in V_g$, where $V_g$ is defined in (3). For any polygonal partition $\mathcal T_n$ of $\Omega$, there exists $u_I\in V^{\Delta ,p}_{n,g}$, with $V^{\Delta ,p}_{n,g}$ introduced in (16), such that

$$\begin{aligned} \vert u- u_I\vert _{1,\mathcal T_n} \le 2 \left| u- q^\Delta _p\right| _{1,\mathcal T_n}\quad \forall q^\Delta _p\in \mathcal {S}^{p,\Delta ,-1}(\mathcal T_n), \end{aligned}$$

where $\mathcal {S}^{p,\Delta ,-1}(\mathcal T_n)$ is the space of discontinuous piecewise harmonic polynomials of degree at most p, that is,

$$\begin{aligned} \mathcal {S}^{p,\Delta ,-1}(\mathcal T_n):= \left\{ q \in L^2(\Omega ): \, q_{|_K} \in \mathbb H_p(K) \ \forall K\in \mathcal T_n\right\} . \end{aligned}$$

(17)

Proof

Define $u_I\in V^{\Delta ,p}_{n,g}$ by

$$\begin{aligned} \int _e(u- u_I) q^e_{p-1}\, d s= 0\quad \forall q^e_{p-1}\in \mathbb P_{p-1}(e),\ \forall e\in \mathcal E_n, \end{aligned}$$

(18)

that is, we fix the degrees of freedom (12) of $u_I$ to be equal to the values of the same functionals applied to the solution $u$. Having this, it holds

$$\begin{aligned} \vert u- u_I\vert _{1,\mathcal T_n} \le \left| u- q^\Delta _p\right| _{1,\mathcal T_n} + \left| u_I- q^\Delta _p\right| _{1,\mathcal T_n} \quad \forall q^\Delta _p\in \mathcal {S}^{p,\Delta ,-1}(\mathcal T_n), \end{aligned}$$

(19)

where $\mathcal {S}^{p,\Delta ,-1}(\mathcal T_n)$ is defined in (17). We focus on the second term on the right-hand side of (19). By integrating by parts and using (18), together with the definition of the space (16), we get

$$\begin{aligned} \begin{aligned} \left| u_I- q^\Delta _p\right| ^2_{1,\mathcal T_n}&= \sum _{K\in \mathcal T_n} \left| u_I- q^\Delta _p\right| ^2_{1,K} \\&= \sum _{K\in \mathcal T_n} \left\{ \int _K\left( u_I- q^\Delta _p\right) \left( \underbrace{-\Delta \left( u_I- q^\Delta _p\right) }_{=0}\right) \, d x+ \sum _{e\in \mathcal E^K}\int _e\left( u_I-q^\Delta _p\right) \, \nabla \left( u_I-q^\Delta _p\right) \cdot \mathbf n_K \, d s\right\} \\&= \sum _{K\in \mathcal T_n} \sum _{e\in \mathcal E^K} \int _e\left( u- q^\Delta _p\right) \, \nabla \left( u_I-q^\Delta _p\right) \cdot \mathbf n_K \, d s. \end{aligned} \end{aligned}$$

(20)

By expanding the right-hand side of (20) and using the Cauchy–Schwarz inequality, we obtain

$$\begin{aligned} \begin{aligned} \left| u_I- q^\Delta _p\right| _{1,\mathcal T_n}^2&= \sum _{K\in \mathcal T_n} \int _{K} \nabla \left( u-q^\Delta _p\right) \cdot \nabla \left( u_I{-} q^\Delta _p\right) \, d x+ \int _K\left( u-q^\Delta _p\right) \underbrace{\Delta \left( u_I-q^\Delta _p\right) }_{=0} \, d x\\&\le \left| u- q^\Delta _p\right| _{1,\mathcal T_n} \left| u_I- q^\Delta _p\right| _{1,\mathcal T_n}. \end{aligned} \end{aligned}$$

Inserting this into (19) gives the result. $\square$

We remark that, with a similar proof of that of Proposition 3.1, one can prove an equivalent result for the non-conforming (non-harmonic) VE spaces of [7]; see Proposition 3.8 below.

3.3 Discrete Bilinear Forms

In this section, we complete the definition of the method (10) by introducing a suitable bilinear form $a_n(\cdot ,\cdot )$, which is explicitly computable. We follow here the typical VEM gospel [13, 18, 36]. It is important to highlight that the local bilinear forms $a^K(\cdot ,\cdot )$ defined in (4) are not explicitly computable on the whole discrete spaces since an explicit representation of functions in the harmonic VE spaces is not available in closed form.

Therefore, we aim at introducing explicit computable discrete bilinear forms $a_n^K(\cdot ,\cdot )$ which mimic their continuous counterparts $a^K(\cdot ,\cdot )$. To this purpose, we observe that the Pythagorean theorem yields

$$\begin{aligned} \begin{aligned}&a^K\left( u_n,v_n\right) = a^K\left( \Pi ^{\nabla ,K}_pu_n, \Pi ^{\nabla ,K}_pv_n\right) + a^K\left( \left( I-\Pi ^{\nabla ,K}_p\right) u_n, \left( I-\Pi ^{\nabla ,K}_p\right) v_n\right) \\&\quad \forall u_n,\,v_n\in V^\Delta (K), \end{aligned} \end{aligned}$$

(21)

where we recall that $\Pi ^{\nabla ,K}_p$ is defined in (15). The first term on the right-hand side of (21) is computable, whereas the second is not. Thus, following [36] and the references therein, we replace this term by a computable symmetric bilinear form $S^K:\ker (\Pi ^{\nabla ,K}_p)\times \ker (\Pi ^{\nabla ,K}_p)\rightarrow \mathbb R$, such that

$$\begin{aligned} c_*(p) \vert v_n\vert ^2_{1,K} \le S^K\left( v_n,v_n\right) \le c^*(p) \vert v_n\vert ^2_{1,K}\quad \forall v_n\in \ker \left( \Pi ^{\nabla ,K}_p\right) , \end{aligned}$$

(22)

where $c_*(p)$ and $c^*(p)$ are two positive constants which may depend on $p$, but are independent of $K$ and, in particular, of $h_K$.

Hence, depending on the choice of the stabilization, a class of candidates for the local discrete symmetric bilinear forms is

$$\begin{aligned} \begin{aligned}&a_n^K\left( u_n,v_n\right) = a^K\left( \Pi ^{\nabla ,K}_pu_n, \Pi ^{\nabla ,K}_pv_n\right) + S^K\left( \left( I-\Pi ^{\nabla ,K}_p\right) u_n, \left( I-\Pi ^{\nabla ,K}_p\right) v_n\right) \\&\quad \forall u_n,\,v_n\in V^\Delta (K). \end{aligned} \end{aligned}$$

(23)

The forms $a_n^K(\cdot ,\cdot )$ satisfy the two following properties:

(P1):

$p$-harmonic consistency: for all $K\in \mathcal T_n$ and for all $p\in \mathbb N$,

$$\begin{aligned} a^K\left( q^\Delta _p, v_n\right) = a_n^K\left( q^\Delta _p, v_n\right) \quad \forall q^\Delta _p\in \mathbb H_p(K),\, \forall v_n\in V^\Delta (K); \end{aligned}$$

(24)

(P2):

stability: for all $K\in \mathcal T_n$ and for all $p\in \mathbb N$,

$$\begin{aligned} \alpha _*(p) \vert v_n\vert ^2_{1,K} \le a_n^K(v_n,v_n) \le \alpha ^*(p) \vert v_n\vert ^2_{1,K} \quad \forall v_n\in V^\Delta (K), \end{aligned}$$

(25)

where $\alpha _*(p) =\min (1,c_*(p))$ and $\alpha ^*(p) =\max (1,c^*(p))$.

Owing to property (P1), $p$ can be addressed to as degree of accuracy of the method, since whenever either of its two entries is a harmonic polynomial of degree $p$, the local discrete bilinear form can be computed exactly, up to machine precision. Moreover, since $a_n^K(\cdot ,\cdot )$ is assumed to be symmetric, (P2) implies continuity

$$\begin{aligned} a_n^K(u_n,v_n) \le \left( a_n^K(u_n,u_n) \right) ^{1/2} \!\!\left( a_n^K(v_n,v_n) \right) ^{1/2} \!\le \alpha ^*(p) \vert u_n\vert _{1,K} \vert v_n\vert _{1,K} \quad \forall u_n, v_n{\in } V^\Delta (K). \end{aligned}$$

(26)

The global discrete bilinear form is defined as

$$\begin{aligned} a_n(u_n,v_n) = \sum _{K\in \mathcal T_n} a_n^K(u_n,v_n) \quad \forall u_n\in V_{n,g_1}^{\Delta ,p}, \, \forall v_n\in V_{n,g_2}^{\Delta ,p} \end{aligned}$$

(27)

for all $g_1,g_2 \in H^{1/2}(\partial \Omega )$. The remainder of this section is devoted to introduce an explicit stabilization $S^K(\cdot ,\cdot )$ with explicit bounds of the constants $c_*(p)$ and $c^*(p)$.

For all $K\in \mathcal T_n$, we define

$$\begin{aligned} S^K(u_n,v_n) = \sum _{e\in \mathcal E^K} \frac{p}{h_e} \left( \Pi ^{0,e}_{p-1}u_n, \Pi ^{0,e}_{p-1}v_n\right) _{0,e} \quad \forall u_n,\, v_n\in \ker \left( \Pi ^{\nabla ,K}_p\right) . \end{aligned}$$

(28)

For this choice of stabilization forms, the following result holds true.

Theorem 3.2

Assume that (D1) and (D2) hold true. Then, for any $K\in \mathcal T_n$, the stabilization $S^K(\cdot ,\cdot )$ defined in (28) satisfies (22) with the bounds

$$\begin{aligned} c_*(p) \gtrsim p^{-2}, \quad c^*(p) \lesssim {\left\{ \begin{array}{ll} p\left( \frac{\log (p)}{p} \right) ^{\frac{\lambda _K}{2 }} &{} \quad \mathrm{if }\,K\mathrm{is\,convex}\\ p\left( \frac{\log (p)}{p} \right) ^{\frac{\lambda _K}{2 \omega _K} - \varepsilon } &{} \quad \mathrm{otherwise }\\ \end{array}\right. } \end{aligned}$$

(29)

for all $\varepsilon >0$ arbitrarily small, where the hidden constants in (29) are independent of $h$ and $p$, and where $\omega _K\pi$ and $\lambda _K\pi$, with $\omega _K$ and $\lambda _K\in (0,2)$, denote the largest interior and the smallest exterior angles of $K$, respectively.

Proof

We assume, without loss of generality, that $h_K= 1$; the general result follows from a scaling argument.

For any function $v_n$ in $V^\Delta (K)$, we have

$$\begin{aligned} \begin{aligned} \vert v_n\vert ^2_{1,K}&= -\int _K\underbrace{(\Delta v_n)}_{=0}v_n\,d x+ \int _{\partial K}\nabla v_n\cdot \mathbf n_K \, v_n\, d s\\&= \sum _{e\in \mathcal E^K} \int _e\nabla v_n\cdot \mathbf n_K \left( \Pi ^{0,e}_{p-1}v_n\right) \, d s\le \left\| \nabla v_n\cdot \mathbf n_K\right\| _{0,\partial K} \left\| \Pi ^{0,\partial K}_{p-1}v_n\right\| _{0,\partial K} \end{aligned} \end{aligned}$$

(30)

where we have set, with an abuse of notation, $(\Pi ^{0,\partial K}_{p-1}v_n) _{|_e} = \Pi ^{0,e}_{p-1}(v_{n_{|_e}})$.We prove that

$$\begin{aligned} \Vert \nabla v_n\cdot \mathbf n_K\Vert _{0,\partial K} \lesssim p^{\frac{3}{2}} \Vert \nabla v_n\cdot \mathbf n_K\Vert _{-\frac{1}{2},\partial K}. \end{aligned}$$

(31)

To this end, we set, for the sake of simplicity, $r_p:=\nabla v_n\cdot \mathbf n_K$, and consider the case $r_p\ne 0$. One has $r_p\in L^2(\partial K)$ with $r_{p|_e} \in \mathbb P_p(e)$ for all $e \in \mathcal E^{K}$. In general, $r_p \notin H^{1/2}(\partial K)$. Further, we introduce the piecewise bubble function $b_{\partial K} \in H^{1/2}(\partial K)$ defined edgewise as

$$\begin{aligned} \left( b_{\partial K}\right) _{|_e}(\mathbf x):=\left( \beta \circ \phi _e^{-1}\right) (\mathbf x) \quad \forall e \in \mathcal {E}^{K}, \end{aligned}$$

where $\phi _e: [-1,1] \rightarrow e$ is the linear transformation mapping the interval $[-1,1]$ to the edge $e$, and $\beta : [-1,1] \rightarrow [0,1]$ is the 1D quadratic bubble function $\beta (x):=4(1-x^2)$.

From the definition of the $H^{-1/2}(\partial K)$ norm, the fact that $r_p\in L^2(\partial K)$, and $r_p b_{\partial K} \in H^{1/2}(\partial K) \backslash \{0\}$, we have

$$\begin{aligned} ||r_p ||_{-\frac{1}{2},\partial K} =\sup _{\psi \in H^{1/2}(\partial K) \backslash \{0\}} \frac{\left( r_p,\psi \right) _{0,\partial K}}{||\psi ||_{\frac{1}{2},\partial K}} \ge \frac{\left( r_p,r_p b_{\partial K}\right) _{0,\partial K}}{\left||r_p b_{\partial K}\right||_{\frac{1}{2},\partial K}} =\frac{\left||r_p b_{\partial K}^{\frac{1}{2}} \right||^2_{0,\partial K}}{\left||r_p b_{\partial K} \right||_{\frac{1}{2},\partial K}}. \end{aligned}$$

(32)

We have the two following polynomial $p$-inverse inequalities:

$$\begin{aligned} \left||r_p b_{\partial K} \right||_{0,e} \le \left||r_p b_{\partial K}^{\frac{1}{2}} \right||_{0,e}, \quad \left|r_p b_{\partial K} \right|_{1,e} \lesssim p\left||r_p b_{\partial K}^{\frac{1}{2}} \right||_{0,e} \quad \forall e \in \mathcal {E}^K. \end{aligned}$$

(33)

The first one is a direct consequence of the fact that the range of $b_{\partial K}$ is [0, 1], and the second one follows from [12, Lemma 2]. Using (33), summing over all edges $e \in \mathcal {E}^K$, and applying interpolation theory, lead to

$$\begin{aligned} \left||r_p b_{\partial K} \right||_{\frac{1}{2},\partial K} \lesssim p^{\frac{1}{2}} \left||r_p b_{\partial K}^{\frac{1}{2}} \right||_{0,\partial K}, \end{aligned}$$

which, together with (32), gives

$$\begin{aligned} \left||r_p \right||_{-\frac{1}{2},\partial K} \gtrsim p^{-\frac{1}{2}} \left||r_p b_{\partial K}^{\frac{1}{2}} \right||_{0,\partial K} \gtrsim p^{-\frac{3}{2}}\left||r_p \right||_{0,\partial K}, \end{aligned}$$

where in the last inequality, [25, Lemma 4] was used. The bound (31) follows immediately.

From (30) and (31), taking also into account that $\Delta v_n= 0$ in K, we get

$$\begin{aligned} \left| v_n\right| ^2_{1,K} \lesssim p^{\frac{3}{2}} \left\| \nabla v_n\cdot \mathbf n_K\right\| _{-\frac{1}{2},\partial K} \left\| \Pi ^{0,\partial K}_{p-1}v_n\right\| _{0,\partial K} \lesssim p^{\frac{3}{2}} \left| v_n\right| _{1,K} \left\| \Pi ^{0,\partial K}_{p-1}v_n\right\| _{0,\partial K}, \end{aligned}$$

where in the last step we have used a Neumann trace inequality, see e.g. [56, Theorem A.33]. This proves the first inequality of (22) with $c_*(p) \gtrsim p^{-2}$.

In order to prove the second one, we can write

$$\begin{aligned} \left\| \Pi ^{0,\partial K}_{p-1}v_n\right\| _{0,\partial K} \le \left\| v_n\right\| _{0,\partial K} \lesssim \left\| v_n\right\| _{0,K}^{\frac{1}{2}} \left| v_n\right| _{1,K}^{\frac{1}{2}}, \end{aligned}$$

(34)

where we have used the stability of the $L^2$ projection, the multiplicative trace inequality, and the Poincaré  inequality, see [28], which is valid since $v_n\in \ker (\Pi ^{\nabla ,K}_p)$ and thus has zero mean value on $\partial K$, see (15).

Let us bound the first factor on the right-hand side of (34). To this end, we define $\overline{v}_n$ as the average of $v_n$ over the polygon $K$. A triangle inequality yields

$$\begin{aligned} \Vert v_n\Vert _{0,K} \le \Vert v_n- \overline{v}_n\Vert _{0,K} + \Vert \overline{v}_n\Vert _{0,K}. \end{aligned}$$

(35)

Recalling that $v_n$ has zero average over $\partial K$, we have

$$\begin{aligned} \Vert \overline{v}_n\Vert _{0,K} = \vert K\vert ^{\frac{1}{2}} \vert \overline{v}_n\vert = \frac{\vert K\vert ^{\frac{1}{2}}}{\vert \partial K\vert } \left| \int _{\partial K} \overline{v}_n- v_n\, d s\right| . \end{aligned}$$

A Cauchy–Schwarz inequality, together with the multiplicative trace inequality, yields

$$\begin{aligned} \Vert \overline{v}_n\Vert _{0,K} \lesssim \Vert v_n- \overline{v}_n\Vert _{0,K}^{\frac{1}{2}} \vert v_n\vert ^{\frac{1}{2}}_{1,K}. \end{aligned}$$

Inserting this inequality in (35) gives

$$\begin{aligned} \Vert v_n\Vert _{0,K} \lesssim \Vert v_n- \overline{v}_n\Vert _{0,K} + \Vert v_n- \overline{v}_n\Vert _{0,K}^{\frac{1}{2}} \vert v_n\vert ^{\frac{1}{2}}_{1,K}. \end{aligned}$$

(36)

From [36, Lemma 3.2], we have

$$\begin{aligned} \Vert v_n- \overline{v}_n\Vert _{0,K}^{\frac{1}{2}} \lesssim {\left\{ \begin{array}{ll} \left( \frac{\log (p)}{p} \right) ^{\lambda _K} \vert v_n\vert _{1,K} &{}\quad \text {if }\, K\text { is\, convex}\\ \left( \frac{\log (p)}{p} \right) ^{\frac{\lambda _K}{\omega _K} - \varepsilon } \vert v_n\vert _{1,K} &{}\quad \text {otherwise}\\ \end{array}\right. } \end{aligned}$$

for all $\varepsilon >0$ arbitrarily small. Inserting this into (36) gives

$$\begin{aligned} \Vert v_n\Vert _{0,K} \lesssim {\left\{ \begin{array}{ll} \left( \frac{\log (p)}{p} \right) ^{\frac{\lambda _K}{2}} \vert v_n\vert _{1,K} &{}\quad \text {if }\, K\,\text { is\, convex}\\ \left( \frac{\log (p)}{p} \right) ^{\frac{\lambda _K}{2\omega _K} - \varepsilon } \vert v_n\vert _{1,K} &{}\quad \text {otherwise},\\ \end{array}\right. } \end{aligned}$$

which, together with (34), gives (22) with $c^*(p)$ as in (29). $\square$

Owing to (25) and (29) one deduces

$$\begin{aligned} \alpha _*(p) \gtrsim p^{-2},\quad \alpha ^*(p) \lesssim {\left\{ \begin{array}{ll} p\left( \frac{\log (p)}{p} \right) ^{\frac{\lambda _K}{2 }} &{}\quad \text {if } K\text { is convex}\\ p\left( \frac{\log (p)}{p} \right) ^{\frac{\lambda _K}{2 \omega _K} - \varepsilon } &{}\quad \text {otherwise }\\ \end{array}\right. } \end{aligned}$$

for all $\varepsilon >0$ arbitrarily small.

Remark 3

In the conforming harmonic VEM setting [36], the following local stabilization forms were introduced:

$$\begin{aligned} S^K(u_n,v_n) = (u_n,v_n)_{\frac{1}{2}, \partial K} \quad \forall K\in \mathcal T_n. \end{aligned}$$

It was proven that employing such stabilization forms leads to have stability constants $\alpha _*(p)$ and $\alpha ^*(p)$ that are independent of the degree of accuracy $p$. However, in the present non-conforming setting, such a stabilization is not computable, as the Dirichlet traces of functions in the local VE spaces are not available in closed form.

We investigate numerically the behavior of the conditioning of the global VE matrix in terms of the degree of accuracy $p$, when employing the local stabilization forms in (28). In Fig. 1, we plot the condition number for different values of $p$, when computing the global stiffness matrix on a Cartesian mesh, a Voronoi-Lloyd mesh , and an Escher horses mesh, see Fig. 2, and note that it grows algebraically with $p$. We remark that the condition number of standard (non-harmonic) VEM can grow exponentially or algebraically with $p$, depending on the choice of the internal degrees of freedom. This was investigated in [48].

Fig. 1

Condition number for different values of $p$ of the global stiffness matrix obtained with the local stabilization forms in (28). A Cartesian mesh, a Voronoi-Lloyd mesh, and an Escher horses mesh have been considered. We observe algebraic growth of the condition number with $p$ for all the tested meshes

Fig. 2

Different types of meshes: mesh made of squares (left), Voronoi-Llyod mesh (center), and mesh made of Escher horses (right)

3.4 Abstract Error Analysis

Along the lines of [13, 17, 36], we provide in this section an abstract error analysis of the method (10), taking into account the non-conformity of the approximation. To this purpose, we introduce the auxiliary bilinear form

$$\begin{aligned} \mathcal N_n : H^1(\Omega ) \times H^{1,nc}_{0} (\mathcal T_n,p) \rightarrow \mathbb R,\quad \mathcal N_n (u,v) = \sum _{e\in \mathcal E_n} \int _e\nabla u\cdot \llbracket v\rrbracket \, d s. \end{aligned}$$

(37)

The following convergence result holds true.

Theorem 3.3

Assume that (D1) and (D2) hold true and consider the non-conforming harmonic VEM (10) defined by choosing the harmonic VE spaces as in (16) and (11), with level of non-conformity, as well as degree of accuracy, equal to $p$, and by choosing the discrete bilinear form as in (27) and (23), with stabilization form $S^K(\cdot ,\cdot )$ satisfying (22). Then, the method is well-posed and the following bound holds true:

$$\begin{aligned} \vert u- u_n\vert _{1,\mathcal T_n} \le \frac{\alpha ^*(p)}{\alpha _*(p)} \left\{ 6 \inf _{q^\Delta _p\in \mathcal {S}^{p,\Delta ,-1}(\mathcal T_n)}\left| u- q^\Delta _p\right| _{1,\mathcal T_n} + \sup _{v_n\in V^{\Delta ,p}_{n,0}} \frac{\mathcal N_n (u,v_n)}{\vert v_n\vert _{1,\mathcal T_n}} \right\} , \end{aligned}$$

(38)

where we recall that $\mathcal {S}^{p,\Delta ,-1}(\mathcal T_n)$ is defined in (17), $\mathcal N_n (\cdot , \cdot )$ is given in (37), and the stability constants $\alpha _*(p)$ and $\alpha ^*(p)$ are introduced in (25).

Proof

The well-posedness of the method follows directly from (9), (25) and the Lax–Milgram lemma.

For the bound (38), we observe that

$$\begin{aligned} \vert u- u_n\vert _{1,\mathcal T_n} \le \vert u- u_I\vert _{1,\mathcal T_n} + \vert u_n- u_I\vert _{1,\mathcal T_n} \quad \forall u_I\in V^{\Delta ,p}_{n,g}. \end{aligned}$$

We estimate the second term on the right-hand side. Set $\delta _n:= u_n- u_I$. Since $u_n,u_I\in V^{\Delta ,p}_{n,g}$, then $\delta _n\in V^{\Delta ,p}_{n,0}$. Therefore, for all $q^\Delta _p\in \mathcal {S}^{p,\Delta ,-1}(\mathcal T_n)$, using (25), (10) and (24), we have

$$\begin{aligned} \begin{aligned} \vert \delta _n\vert ^2_{1,\mathcal T_n}&= \sum _{K\in \mathcal T_n} \vert \delta _n\vert _{1,K}^2 \le \frac{1}{\alpha _*(p)} \sum _{K\in \mathcal T_n} a_n^K(\delta _n,\delta _n) = -\frac{1}{\alpha _*(p)} \sum _{K\in \mathcal T_n} a_n^K(u_I,\delta _n)\\&= -\frac{1}{\alpha _*(p)} \left\{ \sum _{K\in \mathcal T_n} \left[ a_n^K\left( u_I- q^\Delta _p, \delta _n\right) + a^K\left( q^\Delta _p- u,\delta _n\right) \right] + \sum _{K\in \mathcal T_n} a^K(u, \delta _n) \right\} .\\ \end{aligned} \end{aligned}$$

The last term on the right-hand side can be rewritten in the spirit of non-conforming methods. More precisely, we observe that an integration by parts, the fact that $\Delta u=0$ in every $K\in \mathcal T_n$, and the definition (37), yield

$$\begin{aligned} \sum _{K\in \mathcal T_n} a^K(u, \delta _n) = \sum _{K\in \mathcal T_n} \int _{\partial K} \nabla u\cdot \mathbf n_K\,\delta _n\, d s= \sum _{e\in \mathcal E_n} \int _e\nabla u\cdot \llbracket \delta _n\rrbracket \, d s= \mathcal N_n(u,\delta _n). \end{aligned}$$

This, together with the stability property (25), the triangle and the Cauchy–Schwarz inequalities, gives

$$\begin{aligned} \vert \delta _n\vert ^2_{1,\mathcal T_n}\le \frac{1}{\alpha _*(p)}\left[ \left( \alpha ^*(p) \left( \left| u_I-u \right| _{1,\mathcal T_n}+\left| u- q^\Delta _p\right| _{1,\mathcal T_n}\right) +\left| q^\Delta _p-u \right| _{1,\mathcal T_n}\right) \left| \delta _n\right| _{1,\mathcal T_n} +\mathcal N_n\left( u,\delta _n\right) \right] . \end{aligned}$$

Therefore, using Proposition 3.1 and $\alpha ^*(p)\ge 1$, we obtain

$$\begin{aligned} \begin{aligned} \vert \delta _n\vert _{1,\mathcal T_n}&\le \frac{1}{\alpha _*(p)}\left[ \alpha ^*(p) \left( 2\left| u-q^\Delta _p\right| _{1,\mathcal T_n}+\left| u- q^\Delta _p\right| _{1,\mathcal T_n}\right) +\left| q^\Delta _p-u \right| _{1,\mathcal T_n}+\frac{\mathcal N_n\left( u,\delta _n\right) }{\left| \delta _n\right| _{1,\mathcal T_n}}\right] \\&\le \frac{\alpha ^*(p)}{\alpha _*(p)} \left[ 4\left| u- q^\Delta _p\right| _{1,\mathcal T_n}+\frac{\mathcal N_n\left( u,\delta _n\right) }{\left| \delta _n\right| _{1,\mathcal T_n}} \right] , \end{aligned} \end{aligned}$$

and bound (38) readily follows. $\square$

We refer to the term $\frac{\alpha ^*(p)}{\alpha _*(p)}$ appearing in (38) as pollution factor.

Remark 4

It is interesting to note that the counterpart of Theorem 3.3 in the conforming version of the harmonic VEM in [36] states that the error of the method is bounded, up to a constant times the pollution factor $\frac{\alpha ^*(p)}{\alpha _*(p)}$, by a best approximation error with respect to piecewise discontinuous harmonic polynomials, plus the best approximation error with respect to functions in the global approximation space. In the non-conforming setting of the present paper, however, the latter term is not present, thanks to Proposition 3.1. The additional term here is related to the non-conformity.

3.5 $h$- and $p$-Error Analysis

This section is devoted to the $h$- and $p$-analysis of the method (10) employing non-conforming harmonic VE spaces with degree of non-conformity equal to the degree of accuracy of the method.

For the analysis, we have to discuss how to bound the two terms on the right-hand side of (38) in terms of $h$ and $p$. The first term, i.e., the best approximation error with respect to discontinuous harmonic polynomials, can be dealt with following [50, 51]. In particular, we recall the following result from [50, Theorem 2.9] (see also [51, Chapter II]).

Lemma 3.4

Under the star-shapedness assumption (D2), for a given $K\in \mathcal T_n$, we denote by $\lambda _K\, \pi$, $0<\lambda _K<2$, its smallest exterior angle. Then, for every harmonic function $u$ in $H^{s+1}(K)$, $s \ge 0$, there exists a sequence $\{q^\Delta _p\}_{p\in \mathbb N}$, with $q^\Delta _p\in \mathbb H_p(K)$ for all $p\in \mathbb N$ with $p\ge s-1$, such that

$$\begin{aligned} \left| u- q^\Delta _p\right| _{1,K} \le c h_K^s \left( \frac{\log (p)}{p} \right) ^{\lambda _Ks} \Vert u\Vert _{s+1,K}, \end{aligned}$$

(39)

for some positive constant c depending only on $\rho _2$.

Remark 5

We underline that the p-version approximation of harmonic functions by means of harmonic polynomials has different rates of convergence than that of generic (non-harmonic) functions by means of full polynomials. In particular, from (39), one deduces that, on convex elements, a better convergence rate is achieved (i.e., harmonic functions can be better approximated by polynomials than generic functions, even by considering harmonic polynomials only), while on non-convex elements, the rate of approximation gets worse (i.e., the best approximation of harmonic functions by full polynomials fails to be achieved with harmonic polynomials).

Next, we have to bound the non-conformity term $\mathcal N_n (u,v_n)$ introduced in (37). To this purpose, we use tools of non-conforming methods and $hp$-analysis.

Firstly, we define $\Omega _{ext }$ as an extension of the domain $\Omega$, subordinated to polygonal decompositions. More precisely, let $\widetilde{\mathcal T}_n$ be a triangulation of $\Omega$ which is given by the union of local triangulations $\widetilde{\mathcal T}_n(K)$ over each polygon $K\in \mathcal T_n$ ($\widetilde{\mathcal T}_n$ is nested in $\mathcal T_n$); such local triangulations are obtained by connecting the vertices of $K$ to the center of the ball with respect to which $K$ is star-shaped, see assumption (D2). With each triangle $T\in \widetilde{\mathcal T}_n$, we associate $Q(T)$, a parallelogram obtained by reflecting $T$ with respect to the midpoint of one of its edges, which is arbitrarily fixed. Then, we set

$$\begin{aligned} \Omega _{ext }:= \bigcup _{T\in \widetilde{\mathcal T}_n} Q(T). \end{aligned}$$

(40)

Notice that $\Omega _{ext }$ could coincide with $\Omega$.

The following lemma provides an upper bound for the non-conformity term $\mathcal N_n(u,v_n)$.

Lemma 3.5

Assume that (D1)-(D4) are satisfied. Then, for all $s\ge 1$ and for all $u\in H^{s+1}(\Omega _{ext })$, the following bound holds true:

$$\begin{aligned} \vert \mathcal N_n(u, v_n) \vert \le c\, d^s\frac{h^{\min (s,p)}}{p^{s}} \Vert u\Vert _{s+1,\Omega _{ext }} \vert v_n\vert _{1,\mathcal T_n}\quad \forall v_n\in V^{\Delta ,p}_{n,0}, \end{aligned}$$

where c is a positive constant depending only on $\rho _1$, $\rho _2$, $\rho _3$, and $\Lambda$, and d is a positive constant.

Proof

Without loss of generality, let us assume that $h=1$, so that $\rho _3^{-1}\le h_K\le 1$ for all $K\in \mathcal T_n$, due to the assumption (D4); the general assertion follows from a scaling argument.

First, we observe that, for all $v_n\in V^{\Delta ,p}_{n,0}$, the definition of non-conforming spaces and basic properties of orthogonal projectors yield

$$\begin{aligned} \begin{aligned} \vert \mathcal N_n (u,v_n) \vert&= \left| \sum _{e\in \mathcal E_n} \int _e\nabla u\cdot \llbracket v_n\rrbracket \, d s\right| =\left| \sum _{e\in \mathcal E_n} \int _e\left( \nabla u- \Pi ^{0,e}_{p-1}\left( \nabla u\right) \right) \cdot \left( \llbracket v_n\rrbracket - \Pi ^{0,e}_{p-1}\llbracket v_n\rrbracket \right) \, d s\right| \\&\le \sum _{e\in \mathcal E_n} \left\| \nabla u- \Pi ^{0,e}_{p-1}\left( \nabla u\right) \right\| _{0,e} \left\| \llbracket v_n\rrbracket - \Pi ^{0,e}_{p-1}\llbracket v_n\rrbracket \right\| _{0,e}, \end{aligned} \end{aligned}$$

(41)

where we have denoted by $\Pi ^{0,e}_{p-1}$, with an abuse of notation, the $L^2$ projector onto the vectorial polynomial spaces of degree $p-1$ on e.

In order to estimate the first term on the right-hand side, we proceed as follows. Let us consider $\widetilde{\mathcal T}_n$, the union of the local triangulations $\widetilde{\mathcal T}_n(K)$ of each $K\in \mathcal T_n$ defined as above. The triangulation $\widetilde{\mathcal T}_n$ has the property that each $T\in \widetilde{\mathcal T}_n$ is star-shaped with respect to a ball of radius greater than or equal to $\rho _4 h_T$, where $\rho _4$ is a positive constant and $h_T$ is the diameter of the triangle $T$, see [53]. Let now $e\in \mathcal E_n$ be fixed and $K\in \mathcal T_n$ be a polygon with $e\in \mathcal E^K$. Then,

$$\begin{aligned} \left\| \nabla u- \Pi ^{0,e}_{p-1}(\nabla u) \right\| _{0,e} \le \left\| \nabla u- \Pi ^{0,T}_{p-1}(\nabla u) \right\| _{0,e}, \end{aligned}$$

where $\Pi ^{0,T}_{p-1}$ is the $L^2$ projector onto the space of vectorial polynomials of degree at most $p-1$ over $T$, and $T$ is the triangle in $\widetilde{\mathcal T}_n(K)$ with $e\subset \partial T$ (this inequality holds true because the restriction of $\Pi ^{0,T}_{p-1}(\nabla u)$ to e is a vectorial polynomial of degree $p-1$).

For any $v\in H^2(T)$, due to [35, Theorem 3.1], we have

$$\begin{aligned} \left\| \nabla v - \Pi ^{0,T}_{p-1}(\nabla v)\right\| _{0,e} \le \frac{\sqrt{5} + 1}{\sqrt{2}} p^{-\frac{1}{2}} \vert \nabla v \vert _{1,T}. \end{aligned}$$

(42)

Using that $\Pi ^{0,T}_{p-1}\nabla q_p= \nabla q_p$ for all $q_p\in \mathbb {P}_{p}(T)$, owing to (42), we get

$$\begin{aligned} \left\| \nabla u{-} \Pi ^{0,T}_{p-1}(\nabla u) \right\| _{0,e} = \left\| \left( \nabla \left( u- q_p\right) \right) - \Pi ^{0,T}_{p-1}\left( \nabla \left( u- q_p\right) \right) \right\| _{0,e} \lesssim p^{-\frac{1}{2}} \left| \nabla \left( u- q_p\right) \right| _{1,T}. \end{aligned}$$

Applying now standard $hp$-polynomial approximation results, see e.g. [17, Lemma 5.1], we obtain for every $q_p\in \mathbb P_{p} (T)$,

$$\begin{aligned} \left| \nabla \left( u- q_{p-1}\right) \right| _{1,T} \lesssim d^s p^{-s+1} \vert \nabla u\vert _{s,Q(T)}, \end{aligned}$$

(43)

where d is a positive constant and $Q(T)$ is the parallelogram given by the union of $T$ and its reflection defined above.

Moving to the second term in (41), assuming that $e=\partial T^-\cap \partial T^+$, where $T^{\pm } \in \widetilde{\mathcal T}_n$ and $T^{\pm } \subset K^{\pm }$, we have

$$\begin{aligned} \left\| \llbracket v_n\rrbracket - \Pi ^{0,e}_{p-1}\llbracket v_n\rrbracket \right\| _{0,e} \le \left\| v_{n|_{T^+}} - \Pi _{p-1}^{0,T^+} v_{n|_{T^+}} \right\| _{0,e} + \left\| v_{n|_{T^-}} - \Pi _{p-1}^{0,T^-} v_{n|_{T^-}}\right\| _{0,e}. \end{aligned}$$

Then, applying once again [35, Theorem 3.1], we deduce

$$\begin{aligned} \left\| \llbracket v_n\rrbracket - \Pi ^{0,e}_{p-1}\llbracket v_n\rrbracket \right\| _{0,e} \lesssim p^{-\frac{1}{2}} \left( \vert v_{n|_{T^+}} \vert _{1,T^+} + \vert v_{n|_{T^-}} \vert _{1,T^-} \right) . \end{aligned}$$

By combining the bounds of the two terms on the right-hand side of (41) and the definition of the extended domain $\Omega _{ext }$ in (40), we get the assertion. $\square$

We are now ready to state the main $h$- and $p$-error estimate result.

Theorem 3.6

Let $\{\mathcal T_n\}_{n\in \mathbb N}$ be a sequence of polygonal decompositions satisfying (D1)–(D4). Let $u$ and $u_n$ be the solutions to (2) and (10), respectively; we assume that $u$ is the restriction to $\Omega$ of an $H^{s+1}$, $s\ge 1$, function (which we still denote $u$, with a slight abuse of notation), over $\Omega _{ext }$, where $\Omega _{ext }$ is defined in (40). Then, the following a priori $h$- and $p$-error estimate holds true:

$$\begin{aligned} \vert u- u_n\vert _{1,\mathcal T_n} \le c\, d^s\, \frac{\alpha ^*(p)}{\alpha _*(p)}\, h^{\min (s,p)} \left\{ \left( \frac{\log (p)}{p}\right) ^{\min _{K\in \mathcal T_n}(\lambda _K)\,s} +p^{-s} \right\} \Vert u\Vert _{s+1,\Omega _{ext }}, \end{aligned}$$

where c is a positive constant depending only on $\rho _1$, $\rho _2$, $\rho _3$, and $\Lambda$, d is a positive constant, $\lambda _K\, \pi$ denotes the smallest exterior angle of $K$ for each $K\in \mathcal T_n$, and $\frac{\alpha ^*(p)}{\alpha _*(p)}$ is the pollution factor appearing in (38), which is related to the choice of the stabilization.

Proof

It is enough to combine Theorem 3.3 with Lemmata 3.4 and 3.5. $\square$

Assuming, moreover, that $u$, the solution to the problem (2), is the restriction to $\Omega$ of an analytic function defined over $\Omega _{ext }$, where $\Omega _{ext }$ was introduced in (40), it is possible to prove the following result.

Theorem 3.7

Let (D1)–(D4) be valid and assume that $u$, the solution to the problem (2), is the restriction to $\Omega$ of an analytic function defined over $\Omega _{ext }$, given in (40). Then, the following a priori $p$-error estimate holds true:

$$\begin{aligned} \vert u- u_n\vert _ {1,\mathcal T_n} \le c \exp {(-b\,p)}, \end{aligned}$$

for some positive constants b and c, depending again only on $\rho _1$, $\rho _2$, $\rho _3$, $\rho _4$ and $\Lambda$.

Proof

It is enough to use Theorem 3.6, in combination with the tools employed in [17, Theorem 5.2]. $\square$

Remark 6

The construction involving the collection of parallelograms in (40) is instrumental for proving Theorem 3.7. In order to derive the bound of Theorem 3.7 from that of Theorem 3.6, one needs to know the explicit dependence on s of the constant in the bound of Theorem 3.6. This comes at the price of involving the extended domain $\Omega _{ext }$. If one were interested in approximating solutions with finite Sobolev regularity, then there would be no need of employing the construction with the parellelograms Q(T). In particular, equation (43) would be valid also with the norm over the triangle T, instead of over Q(T), on the right-hand side. As a consequence, the bounds in Lemma 3.5 and in Theorem 3.6 would be valid also with the norm of u over $\Omega$, instead of over $\Omega _{ext }$, on the right-hand sides. See [17] for additional details on the $hp$-version in the case of the standard VEM setting.

3.6 Error Estimates in the $L^2$ Norm

This section is devoted to bound the $L^2$ error of method (10) in terms of the energy error and best approximation error with respect to piecewise discontinuous harmonic polynomials. For simplicity, we restrict ourselves to the case of convex domains and of sequences of convex polygons; the non-convex case is discussed in Remark 7.

To this purpose, we firstly recall the definition of non-conforming VE spaces introduced in [7] for the approximation of the Poisson problem, and then we prove $hp$-best approximation estimates by functions in those spaces. The obtained results will be instrumental for proving $L^2$ error estimates for method (10). Throughout the whole section, we assume that p, the parameter used in the definition of local spaces (11), is equal to $k$, the non-conformity parameter, appearing in the definition of the global non-conforming VE space (7).

Let $K\in \mathcal T_n$. We define, for $p\in \mathbb N$ arbitrary,

$$\begin{aligned} V(K):= \left\{ v_n\in H^1(K) \, \mid \, \Delta v_n\in \mathbb P_{p-2}(K),\, \left( \nabla v_n\cdot \mathbf n_{K}\right) _{|_e} \in \mathbb P_{p-1}(e) \ \forall e\in \mathcal E_n \right\} . \end{aligned}$$

It is proved in [7, Lemma 3.1] that the following is a set of degrees of freedom for the space $V(K)$. Given $v_n\in V(K)$, we associate the edge moments defined in (12)

$$\begin{aligned} \frac{1}{h_e} \int _ev_nm_{\alpha }^e\, d s, \quad \forall \alpha =0,\dots , p-1, \, \forall e\in \mathcal E^K, \end{aligned}$$

(44)

plus the bulk moments of the form

$$\begin{aligned} \frac{1}{\vert K\vert } \int _Kv_nm_{\varvec{\alpha }} \, d x, \quad \forall \vert \varvec{\alpha }\vert =0,\dots , p-2 , \end{aligned}$$

(45)

where $\{m_{\varvec{\alpha }}\}_{\vert \varvec{\alpha }\vert =0}^{p-2}$ is any basis of $\mathbb P_{p-2}(K)$.

For all $g\in H^{1/2}(\partial \Omega )$, the global non-conforming spaces in (7) are defined as in the harmonic case:

$$\begin{aligned} V^{k}_{n,g}:= \left\{ v_n\in H^{1,nc}_g(\mathcal T_n,k) \, \mid \, {v_n}_{|_{K}} \in V(K)\ \forall K\in \mathcal T_n\right\} . \end{aligned}$$

(46)

The set of global degrees of freedom is obtained by a standard non-conforming coupling of the local counterparts. The precise treatment of Dirichlet boundary conditions should be dealt with as in Remark 2.

We want to show that, in the $H^1$ seminorm, the error between a regular target function and its interpolant in the space $V^{p}_{n,g}$ defined in (46) can be bounded by the best approximation error in the space of piecewise discontinuous polynomials of degree at most $p$. Notice that neither the convexity of $\Omega$ nor the convexity of the elements are needed here.

Proposition 3.8

Given $g\in H^{1/2}(\partial \Omega )$, let $\psi \in V_g$, where $V_g$ is defined in (3). For every polygonal partition $\mathcal T_n$ of $\Omega$, there exists $\psi _I\in V^{p}_{n,g}$, with $V^{p}_{n,g}$ given in (46), such that

$$\begin{aligned} \left| \psi - \psi _I\right| _{1,\mathcal T_n} \le 2 \left| \psi -q_p\right| _{1,\mathcal T_n} \quad \forall q_p \in \mathcal {S}^{p,-1}(\mathcal T_n), \end{aligned}$$

where $\mathcal {S}^{p,-1}(\mathcal T_n)$ is the space of piecewise discontinuous polynomials, that is,

$$\begin{aligned} \mathcal {S}^{p,-1}\left( \mathcal T_n\right) := \left\{ q \in L^2(\Omega ): \, q_{|_K} \in \mathbb P_p(K) \, \forall K\in \mathcal T_n\right\} . \end{aligned}$$

(47)

Proof

The proof follows the lines of that of Proposition 3.1. Given $\psi \in V_g$, we define $\psi _I\in V^{p}_{n,g}$ by imposing its degrees of freedom as follows:

$$\begin{aligned} \begin{aligned} \frac{1}{h_e} \int _e(\psi _I- \psi ) q_{p-1}^e\, d s&= 0\quad \forall q_{p-1}^e\in \mathbb P_{p-1}(e),\, \forall e\in \mathcal E^K,\, \forall K\in \mathcal T_n.\\ \frac{1}{\vert K\vert } \int _K(\psi _I- \psi ) q_{p-2} \, d x&= 0\quad \forall q_{p-2} \in \mathbb P_{p-2}(K),\, \forall K\in \mathcal T_n,\\ \end{aligned} \end{aligned}$$

(48)

It is important to note that, since the degrees of freedom (44) and (45) are unisolvent for the space $V^{p}_{n,g}$, the interpolant $\psi _I$ is defined in a unique way. Having this, we observe that, for all $K\in \mathcal T_n$,

$$\begin{aligned} \vert \psi - \psi _I\vert _{1,K} \le \vert \psi - q_p\vert _{1,K} + \vert \psi _I- q_p\vert _{1,K}\quad \forall q_p\in \mathbb P_p(K). \end{aligned}$$

We focus on the second term. By integration by parts we get

$$\begin{aligned} \begin{aligned} \vert \psi _I- q_p\vert ^2_{1,K}&= \int _K- \Delta (\psi _I- q_p) (\psi _I- q_p) \, d x+ \int _{\partial K} \nabla (\psi _I-q_p)\cdot \mathbf n_K \, (\psi _I-q_p) \, d s\\&= \int _K- \Delta (\psi _I- q_p) (\psi - q_p) \, d x+ \int _{\partial K} \nabla (\psi _I-q_p)\cdot \mathbf n_K \, (\psi -q_p) \, d s,\\ \end{aligned} \end{aligned}$$

where, in the last identity, the definition of the non-conforming space $V^{p}_{n,g}$, given in (46) and the definition (48) of $\psi _I$ via the degrees of freedom were used. Integrating by parts back, we obtain

$$\begin{aligned} \vert \psi _I- q_p\vert ^2_{1,K} = \int _K\nabla (\psi _I-q_p) \cdot \nabla (\psi - q_p) \, d x\le \vert \psi _I- q_p\vert _{1,K} \vert \psi - q_p\vert _{1,K}. \end{aligned}$$

This concludes the proof. $\square$

We are now ready to prove a bound of the $L^2$ error of the method. We will assume henceforth that $\Omega$ is a convex domain split into a collection of convex polygons. An analogous analysis could be performed in the non-convex case, and slightly worse error estimates could be proven, see Remark 7. Nonetheless, here we stick to the convex setting, since we deem it is clearer.

Theorem 3.9

Let $\Omega$ be a polygonal convex domain and let $\{\mathcal T_n\}_{n\in \mathbb N}$ be a sequence of decompositions into convex polygons satisfying (D1)–(D4). Let $u$ and $u_n$ be the solutions to (2) and (10), respectively; we assume that $u$ is the restriction to $\Omega$ of a $H^{s+1}$, $s\ge 1$, function (which we still denote, with a slight abuse of notation, $u$) over $\Omega _{ext }$, where $\Omega _{ext }$ is defined in (40). Then,

$$\begin{aligned} \begin{aligned} \Vert u- u_n\Vert _{0,\Omega }&\le c\, \left\{ \frac{h^{\min (s,p)+1}}{p^{s+1}} \Vert u\Vert _{s+1,\Omega _{ext }} \right. \\&\left. \quad + \max \left( \frac{h}{p}, h\,\alpha ^*(p) \left( \frac{\log (p)}{p} \right) ^{\max _{K\in \mathcal T_n}\lambda _K} \right) \left( \vert u- u_n\vert _{1,\mathcal T_n} + \inf _{q^\Delta _p\in \mathcal {S}^{p,\Delta ,-1}(\mathcal T_n)}\left| u- q^\Delta _p\right| _{1,\mathcal T_n} \right) \right\} ,\\ \end{aligned} \end{aligned}$$

where c is a positive constant depending only on $\rho _1$, $\rho _2$, $\rho _3$, $\rho _4$ and $\Lambda$, $\alpha ^*(p)$ is the “upper” stability constant appearing in (25), $\mathcal {S}^{p,\Delta ,-1}(\mathcal T_n)$ is defined in (17), and $\lambda _K\pi$ denotes the smallest exterior angle of $K$ for each $K\in \mathcal T_n$.

Proof

We consider the following dual problem: Find $\psi \in H^1(\Omega )$ such that

(49)

Standard stability and a priori regularity theory implies that $\psi \in H^2(\Omega )$ and

$$\begin{aligned} \Vert \psi \Vert _{2,\Omega } \lesssim \Vert u- u_n\Vert _{0,\Omega }, \end{aligned}$$

(50)

where the hidden constant depends only on the domain $\Omega$, see e.g. [44, Theorem 3.2.1.2].

Using (49) and (37), and taking into account that $u- u_n\in H^{1,nc}_{0} (\mathcal T_n,p)$, we obtain the following equivalent expression for the $L^2$ error:

$$\begin{aligned} \Vert u- u_n\Vert ^2_{0,\Omega }&= \sum _{K\in \mathcal T_n} \int _K(-\Delta \psi ) (u- u_n) \, d x\nonumber \\&= \sum _{K\in \mathcal T_n} \left\{ \int _K\nabla \psi \cdot \nabla (u- u_n) \, d x- \int _{\partial K} \nabla \psi \cdot \mathbf n_K \, (u- u_n) \, d s\right\} \nonumber \\&= \sum _{K\in \mathcal T_n} a^K(\psi - \psi _I, u- u_n) + \sum _{K\in \mathcal T_n} a^K(\psi _I, u- u_n) - \mathcal N_n(\psi , u- u_n) \nonumber \\&=: T_1+T_2+T_3, \end{aligned}$$

(51)

where $\psi _I$ is the (unique) function in $V^{p}_{n,0}$, the enlarged space of functions with zero Dirichlet traces introduced in (46), defined from $\psi$ via (48); in particular, $\psi _I$ is not piecewise harmonic, in general.

We begin by bounding term $T_1$. Owing to the Cauchy–Schwarz inequality and Proposition 3.8, we have

$$\begin{aligned} |T_1| \le \vert \psi - \psi _I\vert _{1,\mathcal T_n} \vert u- u_n\vert _{1,\mathcal T_n} \le 2 \vert \psi - q_p\vert _{1,\mathcal T_n} \vert u- u_n\vert _{1,\mathcal T_n} \quad \forall q_p\in \mathcal {S}^{p,-1}(\mathcal T_n), \end{aligned}$$

where $\mathcal {S}^{p,-1}(\mathcal T_n)$ is the space of piecewise discontinuous polynomials introduced in (47). By taking $q_p$ equal to the best approximation of $\psi$ in $\mathcal {S}^{p,-1}(\mathcal T_n)$ and using [17, Lemma 4.2], together with (50), we have

$$\begin{aligned} |T_1| \lesssim \frac{h}{p} \Vert \psi \Vert _{2,\Omega } \vert u- u_n\vert _{1,\mathcal T_n} \lesssim \frac{h}{p} \Vert u- u_n\Vert _{0,\Omega } \vert u-u_n\vert _{1,\mathcal T_n}. \end{aligned}$$

Next, we focus on term $T_3$ on the right-hand side of (51). Following the same steps as in the proof of Lemma 3.5, we obtain

$$\begin{aligned} \begin{aligned} |T_3| = |\mathcal N_n(\psi , u- u_n)| \le \sum _{e\in \mathcal E_n} \left\| \nabla \psi - \Pi ^{0,e}_{p-1}(\nabla \psi ) \right\| _{0,e} \left\| \llbracket u-u_n\rrbracket - \Pi ^{0,e}_{p-1}\llbracket u-u_n\rrbracket \right\| _{0,e},\\ \end{aligned} \end{aligned}$$

where $\Pi ^{0,e}_{p-1}$ denotes here again, with an abuse of notation, the $L^2$ projector onto vectorial polynomial spaces. Applying [35, Theorem 3.1] and [17, Lemma 4.1] similarly as in the proof of Lemma 3.5, together with (50) ($\vert \nabla \psi \vert _{1,K} \le \Vert \psi \Vert _{2,K}$), we get

$$\begin{aligned} \begin{aligned} |T_3| \lesssim \frac{h}{p} \Vert u- u_n\Vert _{0,\Omega } \vert u- u_n\vert _ {1,\mathcal T_n}.\\ \end{aligned} \end{aligned}$$

Finally, we study term $T_2$ on the right-hand side of (51), which can be split as

$$\begin{aligned} T_2 = \sum _{K\in \mathcal T_n} a^K(\psi _I, u- u_n) = \sum _{K\in \mathcal T_n} a^K(u, \psi _I) - \sum _{K\in \mathcal T_n} a^K(u_n, \psi _I) =: T_4+T_5. \end{aligned}$$

(52)

The first term $T_4$ is related to the non-conformity of the discretization spaces, whereas the second term $T_5$ reflects the fact that method (10) does not employ the original bilinear form.

We start to bound term $T_4$. Using computations analogous to those in the proof of Lemma 3.5, it is possible to deduce

$$\begin{aligned} \begin{aligned} |T_4|&= \bigg | \sum _{K\in \mathcal T_n} a^K(u,\psi _I) \bigg | = \bigg | \sum _{K\in \mathcal T_n} \int _{\partial K} \nabla u\cdot \mathbf n_K \, \psi _I\, d s\bigg | = |\mathcal N_n(u,\psi _I) | = |\mathcal N_n(u,\psi _I- \psi )| \\&\le \sum _{e\in \mathcal E_n} \left\| \nabla u - \Pi ^{0,e}_{p-1}(\nabla u) \right\| _{0,e} \left\| \llbracket \psi _I-\psi \rrbracket - \Pi ^{0,e}_{p-1}\llbracket \psi _I-\psi \rrbracket \right\| _{0,e}, \end{aligned} \end{aligned}$$

where in the fourth identity we used the fact that $\mathcal N_n(u,\psi )=0$, which holds since $u$ and $\psi$ are sufficiently regular, and in the last step we used (41). Again, $\Pi ^{0,e}_{p-1}$ has to be understood as the $L^2$ projection onto the vectorial polynomial spaces of degree at most $p-1$ on $e$. Applying [35, Theorem 3.1], Proposition 3.8, [17, Lemma 4.2], and finally (50), leads to

$$\begin{aligned} \begin{aligned} |T_4| \lesssim p^{-1} \vert \nabla \left( u- \Pi ^{\nabla }_pu\right) \vert _{1,\mathcal T_n} \vert \psi -\psi _I\vert _{1,\mathcal T_n} \lesssim \frac{h^{\min (s,p)}}{p^s} \Vert u\Vert _{s+1,\Omega _{ext }} \frac{h}{p} \Vert u- u_n\Vert _{0,\Omega }, \end{aligned} \end{aligned}$$

where we recall that $\Omega _{ext }$ is defined in (40) and where $\Pi ^{\nabla }_p$ is any piecewise energy projector from $H^1(K)$ into $\mathbb P_p(K)$, for all $K\in \mathcal T_n$.

Finally, it remains to treat term $T_5$ on the right-hand side of (52). To this purpose, we consider the following splittings of $\psi$ and $\psi _I$. Firstly, we split $\psi$ into $\psi =\psi ^1 + \psi ^2$, where $\psi ^1$ and $\psi ^2$ are, element by element, solutions to the local problems

(53)

for all $K\in \mathcal T_n$. Using (49), we can also observe that $\psi ^2-\psi$ solve the local problems

Then, (local) standard a priori regularity theory and, afterwards, summation over all elements $K\in \mathcal T_n$ imply the global bound

$$\begin{aligned} \left\| \psi ^2-\psi \right\| _{2,\mathcal T_n} \lesssim \left\| u- u_n\right\| _{0,\Omega }, \end{aligned}$$

(54)

where the broken norm $\Vert \cdot \Vert _{2,\mathcal T_n}$ is defined in (5). With the triangle inequality, (50), and (54), we get

$$\begin{aligned} \left\| \psi ^2 \right\| _{2, \mathcal T_n} \le \left\| \psi - \psi ^2 \right\| _{2,\mathcal T_n} + \left\| \psi \right\| _{2,\Omega } \lesssim \left\| u- u_n\right\| _{0,\Omega }. \end{aligned}$$

(55)

Secondly, we split $\psi _I\in V^{p}_{n,0}$ into $\psi _I= \psi _I^1+\psi _I^2$. We define $\psi _I^2$ as the unique element in $V^{\Delta ,p}_{n,0}$ introduced in (16), which satisfies

$$\begin{aligned} \frac{1}{h_e} \int _e\psi _I^2q_{p-1}^e\, d s= \frac{1}{h_e} \int _e\psi _Iq_{p-1}^e\, d s\quad \forall q_{p-1}^e\in \mathbb P_{p-1}(e), \, \forall e \in {\mathcal {E}}_n. \end{aligned}$$

(56)

Existence and uniqueness of $\psi _I^2$ follow from the fact that we are defining $\psi _I^2$ via unisolvent degrees of freedom for the space $V^{\Delta ,p}_{n,0}$. Owing to (56), the definition of $\psi _I$ in (48), and (53), we deduce

$$\begin{aligned} \frac{1}{h_e} \int _e\psi _I^2q_{p-1}^e\, d s= & {} \frac{1}{h_e} \int _e\psi _Iq_{p-1}^e\, d s= \frac{1}{h_e} \int _e\psi q_{p-1}^e\, d s\\= & {} \frac{1}{h_e} \int _e\psi ^2q_{p-1}^e\, d s\quad \forall q_{p-1}^e\in \mathbb P_{p-1}(e),\, \forall e\in \mathcal E_n. \end{aligned}$$

This entails that $\psi _I^2$ approximates $\psi ^2$ in the sense of Proposition 3.1. Having this, the function $\psi _I^1=\psi _I- \psi _I^2\in V^{p}_{n,0}$ satisfies

Moreover, since $u_n\in V^{\Delta ,k}_{n,g}$, $\psi _I^1$ has the essential feature that it satisfies

$$\begin{aligned} a^K\left( u_n,\psi _I^1\right) = \int _K\left( \underbrace{-\Delta u_n}_{=0}\right) \psi _I^1 \, d x+ \underbrace{\int _{\partial K} \left( \nabla u_n\cdot \mathbf n_{K}\right) \psi _I^1 \, d s}_{=0} = 0. \end{aligned}$$

(57)

We have now all the tools for bounding term $T_5$. Using (57), (10), and (24), we get

$$\begin{aligned} \begin{aligned} |T_5|&= \left| \sum _{K\in \mathcal T_n} a^K\left( u_n,\psi _I^2\right) \right| = \left| \sum _{K\in \mathcal T_n} \left\{ a_n^K\left( u_n,\psi _I^2\right) - a^K\left( u_n, \psi _I^2\right) \right\} \right| \\&= \left| \sum _{K\in \mathcal T_n} \left\{ a_n^K\left( u_n-q^\Delta _p, \psi _I^2-\widetilde{q}^\Delta _p\right) - a^K\left( u_n-q^\Delta _p, \psi _I^2-\widetilde{q}^\Delta _p\right) \right\} \right| \\ \nonumber&\quad \forall q^\Delta _p,\, \widetilde{q}^\Delta _p\in \mathcal {S}^{p,\Delta ,-1}(\mathcal T_n), \end{aligned} \end{aligned}$$

where we recall that $\mathcal {S}^{p,\Delta ,-1}(\mathcal T_n)$ is defined in (17). It is important to highlight that it is in fact a key point of the error analysis to have piecewise harmonic functions in both entries of the discrete bilinear form. By applying the continuity property (26) and the Cauchy–Schwarz inequality, then the triangle inequality and Proposition 3.1, we deduce

$$\begin{aligned} |T_5|&\lesssim \alpha ^*(p) \left| u_n- q^\Delta _p\right| _{1,\mathcal T_n}\left| \psi _I^2 - \widetilde{q}^\Delta _p\right| _{1,\mathcal T_n} \\&\le \alpha ^*(p) \left( \left| u- u_n\right| _{1,\mathcal T_n} + \left| u- q^\Delta _p\right| _{1,\mathcal T_n}\right) \left( \left| \psi ^2 - \psi _I^2 \right| _{1,\mathcal T_n} + \left| \psi ^2 - \widetilde{q}^\Delta _p\right| _{1,\mathcal T_n}\right) \\&\lesssim \alpha ^*(p) \left( \left| u- u_n\right| _{1,\mathcal T_n} + \left| u- q^\Delta _p\right| _{1,\mathcal T_n}\right) \left| \psi ^2 - \widetilde{q}^\Delta _p\right| _{1,\mathcal T_n}. \end{aligned}$$

Thanks to Lemma 3.4 (here, $s=1$) and the bound (55), we have

$$\begin{aligned} \begin{aligned} |T_5|&\lesssim \alpha ^*(p) \left( \left| u- u_n\right| _{1,\mathcal T_n} + \left| u- q^\Delta _p\right| _{1,\mathcal T_n}\right) \,h\left( \frac{\log (p)}{p}\right) ^{\min _{K\in \mathcal T_n} \lambda _K} \left( \sum _{K\in \mathcal T_n} \left\| \psi ^2 \right\| ^2_{2,K} \right) ^{\frac{1}{2}}\\&\lesssim \alpha ^*(p) \left( \left| u- u_n\right| _{1,\mathcal T_n} + \left| u- q^\Delta _p\right| _{1,\mathcal T_n}\right) h\left( \frac{\log (p)}{p} \right) ^{\min _{K\in \mathcal T_n} \lambda _K} \left\| u- u_n\right\| _{0,\Omega }, \end{aligned} \end{aligned}$$

where we recall that, for any $K\in \mathcal T_n$, $\lambda _K\, \pi$ denotes the smallest exterior angle of $K$.

By combining the estimates on all the terms $T_1$ to $T_5$, we get the assertion. $\square$

Remark 7

As already highlighted, the case of non-convex $\Omega$ can be treated analogously. More precisely, given $\omega$ the largest reentrant angle of $\Omega$, the solution of (2) belongs to $H^{1+t}(\Omega )$, with $t=\frac{\pi }{\omega }-\varepsilon$ for all $\varepsilon >0$ arbitrarily small. Standard stability and a priori regularity theory, see [9, Theorem 2.1], gives

$$\begin{aligned} \Vert \psi \Vert _{1+t,\Omega } \le c \Vert u- u_n\Vert _{0,\Omega } \end{aligned}$$

for some positive constant c depending only on the domain $\Omega$. An analogous bound is valid for the counterpart of (54) in the non-convex case. Having this, a straightforward modification of the proof of Theorem 3.9 leads to the $h$- and $p$-error bounds

$$\begin{aligned} \begin{aligned} \Vert u- u_n\Vert _{0,\Omega }&\le \left\{ c\, \frac{h^{\min (s,p)+t}}{p^{s+t}} \Vert u\Vert _{s+1,\Omega _{ext }} \right. \\&\quad + \max \left( \left( \frac{h}{p}\right) ^{t}, \,h^t \,\alpha ^*(p) \left( \frac{\log (p)}{p} \right) ^{\max _{K\in \mathcal T_n}(\lambda _K) \,t} \right) \\&\left. \quad \cdot \left( \vert u- u_n\vert _{1,\mathcal T_n} + \inf _{q^\Delta _p\in \mathcal {S}^{p,\Delta ,-1}(\mathcal T_n)}\vert u- q^\Delta _p\vert _{1,\mathcal T_n} \right) \right\} ,\\ \end{aligned} \end{aligned}$$

where c is a positive constant depending only on the constants $\rho _1$, $\rho _2$, $\rho _3$, $\rho _4$, and $\Lambda$ appearing in (D1)–(D4) and in the proof of Lemma 3.5.

The presence of non-convex polygons in the decomposition $\mathcal T_n$ leads to a possible additional loss in the convergence rate in $p$ of the $L^2$ error, which will depend on the largest interior and exterior angles of the polygons.

3.7 Hints for the Extension to the 3D Case

The aim of this section is to give a hint concerning the extension of what we have presented and discussed so far to the three dimensional case.

Concerning the definition of local harmonic VE spaces, one mimics the strategy suggested in [7] and defines, for every polyhedron $K$ in $\mathbb R^3$ and any fixed $p \in \mathbb N$,

$$\begin{aligned} V^\Delta (K):= \left\{ v_n\in H^1(K) \mid \Delta v_n= 0 \text { in}\, K, \, \left( \nabla v_n\cdot \mathbf n_K\right) _{|_F} \in \mathbb P_{p-1}(F) \ \forall F\text { faces of } K\right\} . \end{aligned}$$

We observe that the definition of the local 3D space is a straightforward extension of its 2D counterpart. We underline that this is not the case when using conforming VEM. In that case, typically, one also requires to have a modified version of the local VE spaces on each face, see [2]. On the one hand, this allows the construction of continuous functions over the boundary of a polyhedron, as well as the construction of projectors onto proper polynomial spaces; on the other, it complicates the $p$-analysis of the method. In the non-conforming framework, however, one does not need to fix any sort of continuity across the interface between faces of a polyhedron and thus it suffices to impose that normal derivatives are polynomials.

The global 3D non-conforming space is built as in the 2D case. Also, the degrees of freedom are given by scaled face moments with respect to polynomials up to order $p-1$.

The abstract definition of the 2D local discrete bilinear form in (23) can also be employed in the 3D case. The (properly scaled) 3D counterpart of the 2D explicit stabilization defined in (28) would be

$$\begin{aligned} S^K(u_n,v_n) = \sum _{F\hbox { faces of}\ K} \frac{p}{h_F} \left( \Pi ^{0,F}_{p-1}u_n, \Pi ^{0,F}_{p-1}v_n\right) _{0,F}, \end{aligned}$$

where, for any face $F$, $\Pi ^{0,F}_{p-1}$ denotes the $L^2$ projector onto $\mathbb P_{p-1}(F)$ of the traces on $F$ of functions in the 3D VE space. Nonetheless, it is not clear whether explicit bounds in terms of $p$ of the stability constants appearing in (22) can be proved for this form. In fact, in the 2D case, $hp$-polynomial inverse estimates in 1D were the key tool for proving Theorem 3.2. In the 3D framework, one needs to employ $hp$-polynomial inverse estimates on general polygons based on weighted norms. We highlight that the approach of [31, Chapter 3], see also [30], could be followed in order to prove such $hp$-weighted inverse inequalities. However, as this extension is quite technical, we do not investigate it here.

Independently of the specific choice of the stabilization, provided that it is symmetric and satisfies (25), the abstract error analysis is dealt with similarly to the 2D case, see Theorem 3.3. The only modification is in the definition of the non-conformity term, which in 3D is defined as

$$\begin{aligned} \mathcal N_n (u,v) = \sum _{F\in \mathcal E_n^3} \int _F\nabla u\cdot \llbracket v\rrbracket _F\, d s\end{aligned}$$

for all conforming functions $u$ and all non-conforming functions $v$, where $\mathcal E_n^3$ denotes the set of faces in the polyhedral decomposition, and $\llbracket \cdot \rrbracket _F$ is defined as in (6) in terms of normal derivatives over faces.

The proof of $h$- and $p$-error bounds for this non-conforming term follows the same lines as in the 2D case, since [35, Theorem 3.1] holds true on simplices in arbitrary space dimension. For the best approximation error, one should use the 3D version of Lemma 3.4, which can be found e.g. in [52, Theorem 3.12].

4 Numerical Results

We present in Sect. 4.1 some numerical tests for the $h$-version and the $p$-version of the method, validating the theoretical results obtained in Sect. 3; we conclude with a discussion and some tests on the $hp$-version in Sect. 4.2. As already mentioned, we refer to “Appendix A” for details on the implementation of the method.

4.1 Numerical Results: $h$- and $p$-Version

In this section, we present numerical experiments validating the theoretical error estimates in the $H^1 (\mathcal T_n)$ ($H^1$, for short) and $L^2$ norms discussed in Theorems 3.6, 3.7, and 3.9.

For the following numerical experiments, we consider boundary value problems of the form (1), on $\Omega :=(0,1)^2$, with known exact solutions given by

• $u_1(x,y)=e^x \sin (y)$,

• $u_2(x,y)=u_2(r,\theta )=r^2 \left( \log (r) \sin (2\theta )+\theta \cos (2\theta ) \right)$.

We underline that $u_1$ is an analytic function in $\Omega$, whereas $u_2 \in H^{3-\epsilon }(\Omega )$ for every $\epsilon >0$ arbitrarily small; moreover, $u_2$ represents the natural singular solution at $\mathbf 0 = (0,0)$ of the Poisson problem on a square domain, see e.g. [9].

We discretize these problems on sequences of quasi-uniform Cartesian meshes and Voronoi-Lloyd meshes of the type shown in Fig. 2, left and center, respectively. We also test on a problem with exact solution $u_1$ on the domain $\Omega$ given by the union of four Escher horses as in Fig. 2, right.

It is important to note that, since an explicit representation of the numerical approximation $u_n$ inside each element is not available, due to the “virtuality” of the basis functions, we cannot compute the $L^2$ and $H^1$ errors of the method directly. Instead, we will compute the following relative errors between $u$ and $\Pi ^{\nabla }_pu_n$, $\Pi ^{\nabla }_p$ being defined in (15):

$$\begin{aligned} \frac{\left||u-\Pi ^{\nabla }_pu_n\right||_{0,\Omega }}{\Vert u\Vert _{0,\Omega }}, \quad \frac{\left||u-\Pi ^{\nabla }_pu_n\right||_{1,\mathcal T_n}}{\Vert u\Vert _{1,\Omega }}. \end{aligned}$$

(58)

We observe that the “computable” $H^1$ error in (58) is related to the exact $H^1$ error. In fact, thanks to Theorem 3.3, we have

$$\begin{aligned} \begin{aligned} \vert u - u_n\vert _{1,\mathcal T_n}&\lesssim \inf _{q^\Delta _p\in \mathcal {S}^{p,\Delta ,-1}(\mathcal T_n)} \left| u- q^\Delta _p\right| _{1,\mathcal T_n} + \sup _{v_n\in V^{\Delta ,p}_{n,0}} \frac{\mathcal N_n (u,v_n)}{\vert v_n\vert _{1,\mathcal T_n}} \\&\le \left| u- \Pi ^{\nabla }_pu_n\right| _{1,\mathcal T_n} + \sup _{v_n\in V^{\Delta ,p}_{n,0}} \frac{\mathcal N_n (u,v_n)}{\vert v_n\vert _{1,\mathcal T_n}}; \end{aligned} \end{aligned}$$

the convergence of the second term on the right-hand side is provided in Lemma 3.5. Moreover, by the triangle inequality and the stability of the $H^1$-projection, one also has

$$\begin{aligned} \begin{aligned} \left| u- \Pi ^{\nabla }_pu_n\right| _{1,\mathcal T_n}&\le \left| u- \Pi ^{\nabla }_pu\right| _{1,\mathcal T_n}+\left| \Pi ^{\nabla }_p(u- u_n) \right| _{1,\mathcal T_n}\\&\le \left| u- \Pi ^{\nabla }_pu\right| _{1,\mathcal T_n}+ \left| u- u_n\right| _{1,\mathcal T_n}; \end{aligned} \end{aligned}$$

the convergence of the second term on the right-hand side is provided in Lemma 3.4.

4.1.1 Numerical Results: $h$-Version

In this section, we verify the algebraic rate of convergence of the $h$-version of the method, validating thus Theorems 3.6 and 3.9 for different degrees of accuracy $p=1,2,3,4,5$ of the method.

The numerical results for the problems in $\Omega = (0,1)^2$ with exact solutions $u_1$ and $u_2$, obtained on sequences of Cartesian and Voronoi-Lloyd meshes, are depicted in Figs. 3 and 4.

Fig. 3

Convergence of the h-version of the method for the analytic solution $u_1$ on quasi-uniform Cartesian (first row) and Voronoi-Lloyd (second row) meshes; relative $H^1$ errors (left) and relative $L^2$ errors (right) defined in (58)

Fig. 4

Convergence of the h-version of the method for the solution $u_2$ with finite Sobolev regularity on quasi-uniform Cartesian (first row) and Voronoi-Lloyd (second row) meshes; relative $H^1$ errors (left) and relative $L^2$ errors (right) defined in (58)

From Theorems 3.6 and 3.9, we expect the $H^1$ and $L^2$ errors to behave like $\mathcal {O}(h^{\min (t,p)})$ and $\mathcal {O}(h^{\min (t,p)+1})$, respectively, where $t+1$ is the regularity of the exact solution u, and p is the degree of accuracy. The numerical results in Figs. 3 and 4 are in agreement with these theoretical estimates. In fact, for $u_1$, which belongs to $H^s(\Omega )$ for all $s \geqslant 0$, we see that the $H^1$ error actually converges with order $\mathcal {O}(h^{p})$, and the $L^2$ error with order $\mathcal {O}(h^{p+1})$ for all degrees of accuracy. On the other hand, we observe convergence rates 1 and 2, respectively, for $p=1$, and convergence rates 2 and 3, respectively, for $p=2,3,4,5$. This is due to the fact that the expected convergence is of order $\mathcal {O}(h^{\min \{2-\epsilon ,p\}})$ in the $H^1$ norm and $\mathcal {O}(h^{\min \{2-\epsilon ,p\}+1})$ in the $L^2$ norm.

4.1.2 Numerical Results: $p$-Version

In this section, we validate the exponential convergence of the $p$-version of the method for the model problem (1) with exact solution $u_1$ on $\Omega =(0,1)^2$ on a Cartesian mesh and a Voronoi mesh made of four elements, respectively, as well as on the domain $\Omega$ given by the union of four Escher horses (see Fig. 2, right). The obtained results are depicted in Fig. 5, where the logarithm of the relative errors defined in (58) is plotted against the polynomial degree p.

Fig. 5

Convergence of the $p$-version of the method for the analytic solution $u_1$ on a quasi-uniform Cartesian mesh, a Voronoi-Lloyd mesh, and a Escher horses mesh; relative $H^1$ errors (left) and relative $L^2$ errors (right) defined in (58)

One can clearly observe that the exponential convergence predicted in Theorem 3.7 is attained, even when employing a very coarse mesh with (non-convex) non-star-shaped elements, as the one in Fig. 2, right.

4.2 The $hp$-Version and Approximation of Corner Singularities

So far, both the theoretical analysis and the numerical tests were performed considering approximation spaces with uniform degree of accuracy $p$ and with quasi-uniform meshes.

In general, however, the solutions to elliptic problems over polygonal domains have natural singularities arising in neighbourhoods of the corners of the domain. In particular, for problem (2) in a domain $\Omega$ with reentrant corners, the solution might have a regularity lower than $H^2$, even if the Dirichlet boundary datum $g$ is smooth; for a precise functional setting regarding regularity of solutions to elliptic PDEs, we refer to [9, 44, 56] and the references therein. This implies that both the $h$- and the $p$-versions of standard Galerkin methods, in general, have limited approximation properties. In particular, employing quasi-uniform meshes and uniform degree of accuracy, does not entail any sort of exponential convergence.

A possible way to recover exponential convergence, even in presence of corner singularities, is to use the so-called $hp$-strategy firstly designed by Babuška and Guo [8,9,10] in the FEM framework, and then generalized to the VEM in [18]. This strategy consists in combining mesh refinement towards the corners of the domain and increasing the number of degrees of freedom over the polygonal decomposition in a non-uniform way. In this section, we discuss and numerically test an hp-version of the presented non-conforming harmonic VEM.

To this purpose, we recall the concept of sequences of geometrically graded polygonal meshes $\{\mathcal T_n\}_{n\in \mathbb N}$. For a given $n \in \mathbb N$, $\mathcal T_n$ is a polygonal mesh consisting of $n+1$ layers, where we define a layer as follows. The so-called 0-th layer is the set of all polygons in $\mathcal T_n$ abutting the vertices of $\Omega$. The other layers are defined inductively by requiring that the $\ell$-th layer consists of those polygons, which abut the polygons in the ($\ell -1$)-th layer. More precisely, for all $\ell =1,\dots ,n$, we set

$$\begin{aligned} L_{n,\ell } := L_\ell := \left\{ K\in \mathcal T_n\mid \overline{K}\cap \overline{K_{\ell -1}} \ne \emptyset \text { for some } K_{\ell -1} \in L_{\ell -1},\, K\not \subseteq \cup _{j=0}^{\ell -1} L_j \right\} . \end{aligned}$$

The $hp$-gospel states that, in order to achieve exponential convergence of the error, one has to employ geometrically graded sequences of meshes. For this reason, we consider sequences $\{\mathcal T_n\}_{n\in \mathbb N}$ satisfying (D1)–(D3), but not (D4); we require instead

(D5):

for all $n\in \mathbb N$, there exists $\sigma \in (0,1)$, called grading parameter, such that

$$\begin{aligned} h_K\approx {\left\{ \begin{array}{ll} \sigma ^{n} &{}\quad \text {if } K\in L_0\\ \frac{1-\sigma }{\sigma } \text {dist}\left( K, \mathcal V^\Omega \right) &{} \quad \text {if } K\in L_\ell ,\, \ell =1,\dots ,n, \end{array}\right. } \end{aligned}$$

(59)

where $\mathcal V^\Omega$ denotes the set of vertices of the polygonal domain $\Omega$.

Sequences $\{\mathcal T_n\}_{n\in \mathbb N}$ satisfying (D5) have the property that the layers “near” the corners of the domain consist of elements with measure converging to zero, whereas the other layers consist of polygons with fixed size. In Fig. 6, we depict three meshes that represent the third elements $\mathcal T_3$ in certain sequences of meshes of the L-shaped domain

$$\begin{aligned} \Omega := (-1,1)^2\setminus (-1,0)^2, \end{aligned}$$

(60)

which are graded, for simplicity, only towards the vertex $\mathbf 0$.

Fig. 6

Third element $\mathcal T_3$ in three different sequences of geometrically graded meshes (type a–c from left to right) with $\sigma =0.5$

We still miss a crucial ingredient for a complete description of the $hp$-strategy, namely harmonic VE spaces with non-uniform degrees of accuracy. For all $n\in \mathbb N$, we can order the elements in $\mathcal T_n$ as $K_1$, $K_2$, ..., $K_{\text {card}(\mathcal T_n)}$; then we consider a vector $\mathbf p_n \in \mathbb N^{\text {card}(\mathcal T_n)}$ whose entries are defined as follows:

$$\begin{aligned} (\mathbf p_n)_j := {\left\{ \begin{array}{ll} 1 &{}\quad \text {if } K_j \in L_0\\ \max (1, \lceil \mu (\ell +1) \rceil ) &{} \quad \text {if } K_j \in L_\ell ,\, \ell = 1,\dots ,n,\\ \end{array}\right. } \end{aligned}$$

(61)

where $\mu$ is a positive parameter to be assigned, and where $\lceil \cdot \rceil$ is the ceiling function.

Having $\mathbf p_n$ for all $n\in \mathbb N$, we consider the elements $e_1$, $e_2$, ..., $e_{\text {card}(\mathcal E_n)}$ in $\mathcal E_n$; we consequently define a vector $\mathbf p^{\mathcal E}_n \in \mathbb N^{\text {card}(\mathcal E_n)}$, whose entries are built using the following rule (maximum rule):

$$\begin{aligned} \left( \mathbf p^{\mathcal E}_n\right) _j := {\left\{ \begin{array}{ll} (\mathbf p_n)_{i} &{} \quad \text {if } e_j \in \mathcal E_n^B \text { and } e_j \subset \partial K_i\\ \max \left( (\mathbf p_n)_{i_1}, (\mathbf p_n)_{i_2}\right) &{}\quad \text {if } e_j \in \mathcal E_n^I \text { and } e_j \subset \partial K_{i_1} \cap \partial K_{i_2}.\\ \end{array}\right. } \end{aligned}$$

At this point, we define the local harmonic VE spaces with non-uniform degrees of accuracy as follows. For all $K\in \mathcal T_n$, we set

$$\begin{aligned} V^\Delta (K):= \left\{ v_n\in H^1{(K)} \, \mid \, \Delta v_n= 0 \text { in } K,\, \left( \nabla v_n\cdot \mathbf n_K\right) _{|_{e_j}} \in \mathbb P_{\left( \mathbf p^{\mathcal E}_n\right) _j}(e_j) \, \forall e_j \text { edge of } K\right\} . \end{aligned}$$

The global non-conforming space and the set of global degrees of freedom are defined similarly to those for the case of uniform degree, see Sect. 3. The difference is that now the degrees of freedom and the corresponding “level of non-conformity” of the method vary from edge to edge. This approach is similar to that discussed in [18] for the $hp$-version of the conforming standard VEM.

Under this construction, one should be able to prove the following convergence result in terms of the number of degrees of freedom. There exists $\mu >0$ such that the choice (61) guarantees

$$\begin{aligned} \vert u- u_n\vert _{1,\mathcal T_n} \le c \exp {\left( -b \root 2 \of {\# \text {dofs} }\right) }, \end{aligned}$$

(62)

for some positive constants b and c, depending on u, $\rho _1$, $\rho _2$, $\Lambda$, and $\sigma$, where $\# \text {dofs}$ denotes the number of degrees of freedom of the discretization space. This exponential convergence in terms of the dimension of the approximation space was proven for conforming harmonic VEM in [36] and for Trefftz DG-FEM in [45]. In the present non-conforming harmonic VEM, the setting of the proof of such exponential convergence would follow the same lines as that of the two methods mentioned above. We omit a detailed analysis and present here some numerical results.

We underline that the exponential convergence in (62) is faster (in terms of the dimension of the space) than that of standard $hp$-FEM [56] and $hp$-VEM [18], whose decay rate is $\mathcal {O}(\exp { \left( -b\root 3 \of {\# \text {dofs}} \right) })$, due to the use of harmonic subspaces instead of complete FE or VE spaces.

For our numerical tests, we consider the boundary value problems (2) on the L-shaped domain $\Omega$ defined in (60), with exact solution

$$\begin{aligned} u_3(x,y)=u_3(r,\theta )=r^{\frac{2}{3}} \sin \left( \frac{2}{3} \theta +\frac{\pi }{3} \right) . \end{aligned}$$

We note that $u_3 \in H^{\frac{5}{3}-\epsilon }(\Omega )$ for every $\epsilon >0$ arbitrarily small, and also $u_3 \in H^{\frac{5}{3}-\epsilon }(\Omega _{ext })$, where $\Omega _{ext }$ is defined in (40); we stress that $u_3$ is the natural solution, singular at $\mathbf 0=(0,0)$, which arises when solving a Poisson problem in the L-shaped domain $\Omega$.

In Fig. 7, we show the convergence of the $hp$-version of the method for different values of the grading parameter $\sigma$ used in (59) and with degrees of accuracy graded according to (61), having set $\mu =1$. We plot the logarithm of the relative $H^1$ error (58) against the square root of the number of degrees of freedom.

Note that, due to the different number of degrees of freedom for each type of mesh, the range of the coordinates varies from plot to plot. The straight lines for $\sigma =0.5$ and $\sigma =\sqrt{2}-1$ indicate agreement with (62) for meshes of type (a) and (b). However, when employing the mesh of type (c) with all grading parameters, and when employing grading parameter $\sigma =(\sqrt{2}-1)^2$ for meshes of all types, we do not observe exponential convergence (62). In the former case, we deem that this is due to the shape of the elements, whereas, in the latter, this could be due to the fact that the size of the elements in the outer layers is too large if picking the parameter $\mu$ in (61) equal to 1.

Fig. 7

Convergence of the hp-version of the method for the solution $u_3$ on an L-shaped domain $\Omega$, for the three sequences of graded meshes represented in Fig. 6; relative $H^1$ errors defined in (58). The grading parameter $\sigma$ is set to 1 / 2, $\sqrt{2}-1$ and $(\sqrt{2}-1)^2$

We point out that, in the framework of the conforming harmonic VEM [36], a similar behaviour for the mesh of type (c) was observed. Instead, when employing the $hp$-version of the standard (non-harmonic) VEM [18], the performance is more robust and the decay of the error is always straight exponential. This suboptimal behaviour might be intrinsic in the use of harmonic polynomials, or might be due to the choice of the harmonic polynomial basis employed in the construction of the method, see “Appendix A”.

5 Conclusions

In this paper, we investigated non-conforming harmonic VEM for the approximation of solutions to 2D Dirichlet–Laplace problems, providing error bounds in terms both of $h$, the mesh size, and of $p$, the degree of accuracy of the method. We gave some hints concerning the extension of the method to the 3D case, where the design of a suitable stabilization is the only missing item. Numerical tests validating the theoretical convergence results, as well as testing the $hp$-version of the method in presence of corner singularities, were presented.

The technology herein introduced can also be seen as an intermediate step towards the construction of non-conforming Trefftz-VE spaces for the approximation of solutions to Helmholtz problems, which has been recently investigated in [49].

Appendix A: Details on the Implementation

In this section, we discuss some practical aspects concerning the implementation of the non-conforming harmonic VEM in 2D. We employ henceforth the notation of [16]. It is worth to underline that we present herein only the case with uniform degree of accuracy; the implementation of the $hp$ version is dealt with similarly. As a first step, we begin by fixing the notation for the various bases instrumental for the construction of the method.

Basis of $\mathbb P_{p-1}(e)$ for a given $e\in \mathcal {E}^K$. Using the same notation as in (12), we denote the basis of $\mathbb P_{p-1}(e)$, $e\in \mathcal E^K$, by $\{ m_{r}^e\}_{r=0,\ldots ,p-1}$. The choice we make is

$$\begin{aligned} m_{r}^e(\mathbf x):=\mathbb L_{r}\left( \phi _e^{-1}(\mathbf x) \right) \quad \forall r=0,\dots ,p-1, \end{aligned}$$

(63)

where $\phi _e: [-1,1] \rightarrow e$ is the linear transformation mapping the interval $[-1,1]$ to the edge $e$, and $\mathbb L_{r}$ is the Legendre polynomial of degree r over $[-1,1]$. We recall, see e.g. [56], for future use the orthogonality property

$$\begin{aligned} \left( m_r^e,m_s^e\right) _{0,e} = \frac{h_e}{2} \int _{-1}^{1} \mathbb L_r(t) \mathbb L_s(t) \, \text {d}t = \frac{h_e}{2r+1} \delta _{rs} \quad \forall r,s=0,\dots ,p-1, \end{aligned}$$

(64)

where $\delta _{rs}$ is the Kronecker delta (1 if $r=s$, 0 otherwise).

Basis of $\mathbb H_p(K)$ for a given $K\in \mathcal T_n$. We denote the basis of the space of harmonic polynomials $\mathbb H_p(K)$ by $\{ q_\alpha ^\Delta \}_{\alpha =1,\ldots ,n_p^\Delta }$, where $n_p^\Delta :=\dim \mathbb H_p(K) = 2p+1$. The choice we make for this basis is

$$\begin{aligned} \begin{aligned} q_1^\Delta (\mathbf x)&= 1;\\ q_{2l}^\Delta (\mathbf x)&= \sum _{k=1, \, k \text { odd }}^{l} (-1)^{\frac{k-1}{2}} \left( {\begin{array}{c}l\\ k\end{array}}\right) \left( \frac{x-x_K}{h_K} \right) ^{l-k} \left( \frac{y-y_K}{h_K} \right) ^{k} \quad \forall l=1,\dots ,p; \\ q_{2l+1}^\Delta (\mathbf x)&= \sum _{k=0, \, k \text { even }}^{l} (-1)^{\frac{k}{2}} \left( {\begin{array}{c}l\\ k\end{array}}\right) \left( \frac{x-x_K}{h_K} \right) ^{l-k} \left( \frac{y-y_K}{h_K} \right) ^{k} \quad \forall l=1,\dots ,p.\\ \end{aligned} \end{aligned}$$

The fact that this is actually a basis for $\mathbb H_p(K)$ is proven, e.g., in [6, Theorem 5.24].

Basis for $V^\Delta (K)$ for a given $K\in \mathcal T_n$. For this local VE space introduced in (11), we employ the canonical basis $\left\{ \varphi _{j,r} \right\} _{\begin{array}{c} j=1,\ldots , N_K\\ r=0,\ldots ,p-1 \end{array}}$ defined though (13), where we also recall that $N_K$ denotes the number of edges of K.

In the following, we derive the matrix representation of the local discrete bilinear form introduced in (23). We begin with the computation of the matrix representation of the projector $\Pi ^{\nabla ,K}_p$ acting from $V(K)$ to $\mathbb H_p(K)$ and defined in (15). To this purpose, given any basis function $\varphi _{j,r} \in V^\Delta (K)$, $j=1,\dots ,N_K$, $r=0,\dots ,p-1$, we expand $\Pi ^{\nabla ,K}_p\varphi _{j,r}$ in terms of basis $\{ q_\alpha ^\Delta \}_{\alpha =1,\ldots ,n_p^\Delta }$ of $\mathbb H_p(K)$, i.e.,

$$\begin{aligned} \Pi ^{\nabla ,K}_p\varphi _{j,r} = \sum _{\alpha =1}^{n_p^\Delta } s_\alpha ^{(j,r)} q_\alpha ^\Delta . \end{aligned}$$

(65)

Using (15) and testing (65) with functions $q_\beta ^\Delta$, $\beta =1,\dots ,n_p^\Delta$, we get that the coefficients $s_{\alpha }^{(j,r)}$ can be computed by solving for $\varvec{s}^{(j,r)}:=[s_1^{(j,r)},\ldots , s_{n_p^\Delta }^{(j,r)}]^T$ the $n_p^\Delta \times n_p^\Delta$ algebraic linear system

$$\begin{aligned} \varvec{G}\varvec{s}^{(j,r)}=\varvec{b}^{(j,r)}, \end{aligned}$$

where

$$\begin{aligned}&\varvec{G}= \begin{bmatrix} \left( q_1^\Delta ,1\right) _{0,\partial K}&\quad \left( q_2^\Delta ,1\right) _{0,\partial K}&\quad \cdots&\quad \left( q_{n_p^\Delta }^\Delta ,1\right) _{0,\partial K} \\ 0&\quad \left( \nabla q_2^\Delta ,\nabla q_2^\Delta \right) _{0,K}&\quad \cdots&\quad \left( \nabla q_{n_p^\Delta }^\Delta ,\nabla q_2^\Delta \right) _{0,K} \\ \vdots&\quad \vdots&\quad \ddots&\quad \vdots \\ 0&\quad \left( \nabla q_{n_p^\Delta }^\Delta ,\nabla q_2^\Delta \right) _{0,K}&\quad \cdots&\quad \left( \nabla q_{n_p^\Delta }^\Delta ,\nabla q_{n_p^\Delta }^\Delta \right) _{0,K} \end{bmatrix}, \nonumber \\&\quad \varvec{b}^{(j,r)}= \begin{bmatrix} \left( \varphi _{j,r},1\right) _{0,\partial K} \\ \left( \nabla \varphi _{j,r},\nabla q_2^\Delta \right) _{0,K} \\ \vdots \\ \left( \nabla \varphi _{j,r},\nabla q_{n_p^\Delta }^\Delta \right) _{0,K} \end{bmatrix}. \end{aligned}$$

Collecting all the $N_K p$ (column) vectors $\varvec{b}^{(j,r)}$ in a matrix $\varvec{B}\in \mathbb R^{n_p^\Delta \times N_K p}$, namely, setting $\varvec{B}:=[\varvec{b}^{(1,1)},\dots ,\varvec{b}^{(N_K,p)}]$, the matrix representation $\varvec{\Pi }_*$ of the projector $\Pi ^{\nabla ,K}_p$ acting from $V^\Delta (K)$ to $\mathbb H_p(K)$ is given by

$$\begin{aligned} \varvec{\Pi }_*= \varvec{G}^{-1} \varvec{B} \in \mathbb R^{n_p^\Delta \times N_K p}. \end{aligned}$$

Subsequently, we define

$$\begin{aligned} \varvec{D}:=\begin{bmatrix} \text {dof}_{1,1}\left( q_1^\Delta \right)&\quad \cdots&\quad \text {dof}_{1,1}\left( q_{n_p^\Delta }^\Delta \right) \\ \vdots&\quad \ddots&\quad \vdots \\ \text {dof}_{N_K, p}\left( q_1^\Delta \right)&\quad \cdots&\quad \text {dof}_{N_K, p}\left( q_{n_p^\Delta }^\Delta \right) \end{bmatrix} \in \mathbb R^{N_Kp \times n_p^\Delta }. \end{aligned}$$

Let $\varvec{\Pi }$ be the matrix representation of the operator $\Pi ^{\nabla ,K}_p$ seen now as a map from $V^\Delta (K)$ into $V^\Delta (K)\supseteq \mathbb H_p(K)$. Then, following [16], it is possible to show that

$$\begin{aligned} \varvec{\Pi }= \varvec{D} \varvec{G}^{-1} \varvec{B} \in \mathbb R^{N_K p \times N_K p}. \end{aligned}$$

Next, denoting by $\widetilde{\varvec{G}} \in \mathbb R^{n_p^\Delta \times n_p^\Delta }$ the matrix coinciding with $\varvec{G}$ apart from the first row which is set to zero, the matrix representation of the bilinear form in (23) is

$$\begin{aligned} \left( \varvec{\Pi }_*\right) ^T \, \widetilde{\varvec{G}} \,\left( \varvec{\Pi }_*\right) + \left( \varvec{I} - \varvec{\Pi }\right) ^T \, \varvec{S} \, \left( \varvec{I} - \varvec{\Pi }\right) . \end{aligned}$$

Here, $\varvec{S}$ denotes the matrix representation of an explicit stabilization $S^K(\cdot ,\cdot )$. For the stabilization defined in (28), we have

$$\begin{aligned}&\varvec{S}((k-1)N_K+ r,(l-1)N_K+s) = \sum _{i=1}^{N_K} \frac{p}{h_{e_i}} \left( \Pi ^{0,e_i}_{p-1}\varphi _{l,s}, \Pi ^{0,e_i}_{p-1}\varphi _{k,r}\right) _{0,e_i}\\&\quad \forall k,l=1,\dots ,N_K, \forall r,s=0,\dots ,p-1. \end{aligned}$$

By expanding $\Pi ^{0,e_i}_{p-1}\varphi _{l,s}$ and $\Pi ^{0,e_i}_{p-1}\varphi _{k,r}$ in the basis $\left\{ m_\gamma ^{e_i} \right\} _{\gamma =0,\ldots ,p-1}$ of $\mathbb P_{p-1}(e_i)$, i.e.,

$$\begin{aligned}&\Pi ^{0,e_i}_{p-1}\varphi _{l,s} = \sum _{\gamma =0}^{p-1} t_\gamma ^{(l,s),e_i} m_\gamma ^{e_i}, \quad \Pi ^{0,e_i}_{p-1}\varphi _{k,r}= \sum _{\zeta =0}^{p-1}t_\zeta ^{(k,r),{e_i}} m_\zeta ^{e_i}, \nonumber \\&\quad \forall k,l=1,\dots ,N_K,\, \forall r,s=0,\dots ,p-1, \end{aligned}$$

(66)

we can write

$$\begin{aligned} \begin{aligned}&\varvec{S}((k-1)N_K+ r,(l-1)N_K+s) = \sum _{i=1}^{N_K} \sum _{\gamma =0}^{p-1} \sum _{\zeta =0}^{p-1} t_\gamma ^{(l,s),{e_i}} t_\zeta ^{(k,r),{e_i}} \frac{p}{h_{e_i}} \left( m_\gamma ^{e_i}, m_\zeta ^{e_i}\right) _{0,{e_i}}\nonumber \\&\quad \forall k,l=1,\dots ,N_K,\, \forall r,s=0,\dots ,p-1. \end{aligned} \end{aligned}$$

For the basis defined in (63), using the orthogonality of the Legendre polynomials (64), this expression can be simplified leading to a diagonal stability matrix $\mathbf S$:

$$\begin{aligned} \begin{aligned}&\varvec{S}((k-1)N_K+ r,(k-1)N_K+r)= \sum _{i=1}^{N_K} \sum _{\zeta =0}^{p-1} \frac{p}{2r+1} \left( t_\zeta ^{(k,r),{e_i}}\right) ^2 \nonumber \\&\quad \forall k=1,\dots ,N_K,\, \forall r=0,\dots ,p-1. \end{aligned} \end{aligned}$$

For fixed $i,k \in \{1,\dots ,N_K\}$ and $r \in \{0,\dots ,p-1\}$, the coefficients $t_\zeta ^{(k,r),e_i}$ are obtained by testing $\Pi ^{0,e_i}_{p-1}\varphi _{k,r}$, defined in (66), with $m_{\zeta }^{e_i}$, $\zeta =0,\dots ,p-1$, and by taking into account the definition of $\Pi ^{0,e_i}_{p-1}$ in (14), the orthogonality relation (64) and the definition of $\varphi _{k,r}$ in (13). This gives

$$\begin{aligned} t_\zeta ^{(k,r),e_i} = \frac{2 \zeta +1}{h_{e_i}} \left( \varphi _{k,r},m_\zeta ^{e_i}\right) _{0,e_i} = (2 \zeta +1) \delta _{i k} \delta _{r \zeta } \quad \forall \zeta =0,\dots ,p-1. \end{aligned}$$

The global system of linear equations corresponding to method (10) is assembled as in the standard non-conforming FEM. Finally, one imposes in a non-conforming fashion the Dirichlet boundary datum $g$ by

$$\begin{aligned} \int _eu_nq^e_{p-1}\, d s= \int _egq^e_{p-1}\, d s\quad \forall q^e_{p-1}\in \mathbb P_{p-1}(e), \end{aligned}$$

where, in practice, g is replaced by $g_p$, see Remark 2.