Pointwise error estimates of linear finite element method for Neumann boundary value problems in a smooth domain

Abstract

Pointwise error analysis of the linear finite element approximation for $-\,\Delta u + u = f$ in $\Omega$, $\partial _n u = \tau$ on $\partial \Omega$, where $\Omega$ is a bounded smooth domain in $\mathbb {R}^N$, is presented. We establish $O(h^2|\log h|)$ and O(h) error bounds in the $L^\infty$- and $W^{1,\infty }$-norms respectively, by adopting the technique of regularized Green’s functions combined with local $H^1$- and $L^2$-estimates in dyadic annuli. Since the computational domain $\Omega _h$ is only polyhedral, one has to take into account non-conformity of the approximation caused by the discrepancy $\Omega _h \ne \Omega$. In particular, the so-called Galerkin orthogonality relation, utilized three times in the proof, does not exactly hold and involves domain perturbation terms (or boundary-skin terms), which need to be addressed carefully. A numerical example is provided to confirm the theoretical result.

1 Introduction

We consider the following Poisson equation with a non-homogeneous Neumann boundary condition:

$$\begin{aligned} -\,\Delta u + u = f \quad \text {in}\quad \Omega , \quad \partial _n u = \tau \quad \text {on}\quad \Gamma := \partial \Omega , \end{aligned}$$

(1.1)

where $\Omega \subset \mathbb {R}^N$ is a bounded domain with a smooth boundary $\Gamma$ of $C^\infty$-class, f is an external force, $\tau$ is a prescribed Neumann data, and $\partial _n$ means the directional derivative with respect to the unit outward normal vector n to $\Gamma$. The linear (or $P_1$) finite element approximation to (1.1) is quite standard. Given an approximate polyhedral domain $\Omega _h$ whose vertices lie on $\Gamma$, one can construct a triangulation $\mathcal {T}_h$ of $\Omega _h$, build a finite dimensional space $V_h$ consisting of piecewise linear functions, and seek for $u_h \in V_h$ such that

$$\begin{aligned} (\nabla u_h, \nabla v_h)_{\Omega _h} + (u_h, v_h)_{\Omega _h} = (\tilde{f}, v_h)_{\Omega _h} + (\tilde{\tau }, v_h)_{\Gamma _h} \quad \forall v_h \in V_h, \end{aligned}$$

(1.2)

where $\Gamma _h := \partial \Omega _h$, and $\tilde{f}$ and $\tilde{\tau }$ denote extensions of f and $\tau$, respectively. Then, the main result of this paper is the following pointwise error estimates in the $L^\infty$- and $W^{1,\infty }$-norms:

$$\begin{aligned} \Vert \tilde{u} - u_h\Vert _{L^\infty (\Omega _h)}\le & {} Ch^2|\log h| \, \Vert u\Vert _{W^{2, \infty }(\Omega )}, \nonumber \\ \Vert \tilde{u} - u_h\Vert _{W^{1,\infty }(\Omega _h)}\le & {} Ch \, \Vert u\Vert _{W^{2, \infty }(\Omega )}, \end{aligned}$$

(1.3)

where h denotes the mesh size of $\mathcal {T}_h$, and $\tilde{u}$ is an arbitrary extension of u (of course, the way of extension must enjoy some stability, cf. Sect.  below).

Regarding pointwise error estimates of the finite element method, there have been many contributions since 1970s (for example, see the references in [14]), and, consequently, standard methods to derive them are now available. The strategy of those methods is briefly explained as follows. By duality, analysis of $L^\infty$- or $W^{1,\infty }$-error of $u - u_h$ may be reduced to that of $W^{1,1}$-error between a regularized Green’s function g, with singularity near $x_0 \in \Omega$, and its finite element approximation $g_h$. To deal with $\Vert \nabla (g - g_h)\Vert _{L^1(\Omega )}$ in terms of energy norms, it is estimated either by $\sum _{j=0}^J d_j^{N/2}\Vert \nabla (g - g_h)\Vert _{L^2(\Omega \cap A_j)}$ or by $\Vert \sigma ^{N/2} \nabla (g - g_h)\Vert _{L^2(\Omega )}$, where $\{d_j\}_{j=0}^J$ are radii of dyadic annuli $A_j$ shrinking to $x_0$ with the minimum $d_J = Kh$, whereas $\sigma (x) := (|x - x_0|^2 + \kappa h^2)^{1/2}$. The two strategies may be regarded as using discrete and continuous weights, respectively, and basically lead to the same results. In this paper, we employ the first approach, in which scaling heuristics seem to work easier (in the second approach one actually needs to introduce an artificial parameter $\lambda \in (0,1)$ to avoid singular integration, which makes the weighted norm slightly complicated, cf. Remark 8.4.4 of [3]).

The main difficulty of our problem lies in the non-conformity $V_h \not \subset H^1(\Omega )$ arising from the discrepancy $\Omega _h \ne \Omega$ and $\Gamma _h\ne \Gamma$, which we refer to as domain perturbation. In fact, the so-called Galerkin orthogonality relation (or consistency) does not exactly hold, and hence the standard methodology of error estimate cannot be directly applied. This issue was already considered in classical literature (see [18, Section 4.4] or [5, Section 4.4]) as long as energy-norm (i.e. $H^1$) error estimates for a Dirichlet problem are concerned. However, there are much fewer studies of error analysis in other norms or for other boundary value problems, which take into account domain perturbation. For example, Barrett and Elliott [2], Čermák [4] gave optimal $L^2$-error estimates for a Robin boundary value problem.

As for pointwise error estimates, the issue of domain perturbation was mainly treated only for a homogeneous Dirichlet problem in a convex domain. In this case, one has a conforming approximation $V_h \cap H^1_0(\Omega _h) \subset H^1_0(\Omega )$ with the aid of the zero extension, which makes error analysis simpler. This situation was studied for elliptic problems in [1, 17] and for parabolic ones in [8, 19]. Although an idea to treat $\Omega _h \not \subset \Omega$ in the case of $L^\infty$-analysis is found in [17, p. 2], it does not seem to be directly applicable to $W^{1,\infty }$-analysis or to Neumann problems. In [8, 14, 16], they considered Neumann problems in a smooth domain assuming that triangulations exactly fit a curved boundary, where one need not take into account domain perturbation. This assumption, however, excludes the use of usual Lagrange finite elements. The $P_2$-isoparametric finite element analysis for a Dirichlet problem $(N=2)$ was shown in [20], where the rate of convergence $O(h^{3-\epsilon })$ in the $L^\infty$-norm was obtained.

The aim of this paper is to present pointwise error analysis of the finite element method taking into account full non-conformity caused by domain perturbation. We emphasize that a rigorous proof of such results for Neumann problems remained open even in the simplest setting, i.e., the linear finite element approximation. Therefore, in the present paper, we focus on showing how the issues of domain perturbation can be managed and confine ourselves to the linear approximation. Our main result (1.3) implies that domain perturbation does not affect the rate of convergence in the $L^\infty$- and $W^{1,\infty }$-norms known for the case $\Omega _h = \Omega$ when $P_1$-elements are used to approximate both a curved domain and a solution. We would like to extend this to higher order cases (e.g. isoparametric finite elements) in future work, by adopting the strategy developed in this paper to manage domain perturbation.

Finally, let us make a comment concerning the opinion that the issue of $\Omega _h \ne \Omega$ is similar to that of numerical integration (see [16, p. 1356]). As mentioned in the same paragraph there, if a computational domain is extended (or transformed) to include $\Omega$ and a restriction (or transformation) operator to $\Omega$ is applied, then one can disregard the effect of domain perturbation (higher-order schemes based on such a strategy are proposed e.g. in [6]). On the other hand, since implementing such a restriction operator precisely for general domains is non-trivial in practical computation, some approximation of geometric information for $\Omega$ should be incorporated in the end. Thereby one needs to more or less deal with domain perturbation in error analysis, and, in our opinion, its rigorous treatment would be quite different from that of numerical integration.

The organization of this paper is as follows. Basic notations are introduced in Sect. , together with boundary-skin estimates and a concept of dyadic decomposition. In Sect. , we present the main result (Theorem ) and reduce its proof to $W^{1,1}$-error estimate of $g - g_h$. The weighted $H^1$- and $L^2$-error estimates of $g - g_h$ are shown in Sects. 4 and 5, respectively, which are then combined to complete the proof of Theorem in Sect. . A numerical example is given to confirm the theoretical result in Sect. . Throughout this paper, $C>0$ will denote generic constants which may be different at each occurrence; its dependency (or independency) on other quantities will often be mentioned as well. However, when it appears with sub- or super-scripts (e.g., $C_{0E}, C'$), we do not treat it as generic.

2 Preliminaries

2.1 Basic notation

Recall that $\Omega \subset \mathbb {R}^N$ is a bounded $C^\infty$-domain. We employ the standard notation of the Lebesgue spaces $L^p(\Omega )$, Sobolev spaces $W^{s,p}(\Omega )$ (in particular, $H^s(\Omega ) := W^{s, 2}(\Omega )$), and Hölder spaces $C^{m,\alpha }(\overline{\Omega })$. Throughout this paper we assume the regularity $u \in W^{2,\infty }(\Omega )$ for (1.1), which is indeed true if $f \in C^\alpha (\overline{\Omega })$ and $\tau \in C^\alpha (\overline{\Gamma })$ for some $\alpha \in (0,1)$.

Given a bounded domain $D \subset \mathbb {R}^N$, both of the N-dimensional Lebesgue measure of D and the $(N-1)$-dimensional surface measure of $\partial D$ are simply denoted by |D| and $|\partial D|$, as far as there is no fear of confusion. Furthermore, we let $(\cdot , \cdot )_D$ and $(\cdot , \cdot )_{\partial D}$ be the $L^2(D)$- and $L^2(\partial D)$-inner products, respectively, and define the bilinear form

$$\begin{aligned} a_D(u, v) := (\nabla u, \nabla v)_D + (u, v)_D, \quad u,v \in H^1(D), \end{aligned}$$

which is simply written as a(u, v) when $D = \Omega$, and as $a_h(u, v)$ when $D = \Omega _h$ (to be defined below).

Letting $\Omega _h$ be a polyhedral domain, we consider a family of triangulations $\{\mathcal {T}_h\}_{h\downarrow 0}$ of $\Omega _h$ which consist of closed and mutually disjoint simplices. We assume that $\{\mathcal {T}_h\}_{h\downarrow 0}$ is quasi-uniform, that is, every $T \in \mathcal {T}_h$ contains (resp. is contained in) a ball with the radius ch (resp. h), where $h := \max _{T\in \mathcal {T}_h} h_T$ with $h_T := {\text {diam}}\,T$. The boundary mesh on $\Gamma _h := \partial \Omega _h$ inherited from $\mathcal {T}_h$ is denoted by $\mathcal {S}_h$, namely, $\mathcal {S}_h = \{S \subset \Gamma _h \,|\, S$ is an $(N-1)$-dimensional face of some $T \in \mathcal {T}_h \}$. We then assume that the vertices of every $S \in \mathcal {S}_h$ belong to $\Gamma$, that is, $\Gamma _h$ is essentially a linear interpolation of $\Gamma$.

The linear (or $P_1$) finite element space $V_h$ is given in a standard manner, i.e.,

$$\begin{aligned} V_h = \left\{ v_h \in C(\overline{\Omega }_h){:}\, v_h|_T \in P_1(T) \quad \forall T\in \mathcal {T}_h \right\} , \end{aligned}$$

where $P_k(T)$ stands for the polynomial functions defined in T with degree $\le k$.

Let us recall a well-known result of an interpolation operator (also known as a local regularization operator) $\mathcal {I}_h{:}\,H^1(\Omega _h) \rightarrow V_h$ satisfying the following property (see [3, Section 4.8]):

$$\begin{aligned} \Vert \nabla ^k(v - \mathcal {I}_h v)\Vert _{L^p(T)} \le C_{\mathcal {I}} h_T^{m-k} \Vert \nabla ^m v\Vert _{L^p(M_T)} \quad k=0,1, \, m=1,2, \, v\in W^{m,p}(\Omega _h), \end{aligned}$$

where $M_T := \bigcup \{T' \in \mathcal {T}_h{:}\, T'\cap T \ne \emptyset \}$ is a macro-element of $T \in \mathcal {T}_h$. The constant $C_{\mathcal {I}}$ depends on c, k, m, p and on a reference element; especially it is independent of v and $h_T$. We also use the trace estimate

$$\begin{aligned} \Vert v\Vert _{L^2(\Gamma _h)} \le C \Vert v\Vert _{L^2(\Omega _h)}^{1/2} \Vert v\Vert _{H^1(\Omega _h)}^{1/2}, \end{aligned}$$

where C depends on the $C^{0,1}$-regularity of $\Omega _h$ and thus it is uniformly bounded by that of $\Omega$ for $h\le 1$.

2.2 Boundary-skin estimates

To examine the effects due to the domain discrepancy $\Omega _h \ne \Omega$, we introduce a notion of tubular neighborhoods $\Gamma (\delta ) := \{x\in \mathbb {R}^N{:}\, {\text {dist}}(x, \Gamma ) \le \delta \}$. It is known that (see [9, Section 14.6]) there exists $\delta _0>0$, which depends on the $C^{1,1}$-regularity of $\Omega$, such that each $x \in \Gamma (\delta _0)$ admits a unique representation

$$\begin{aligned} x = \bar{x} + tn(\bar{x}), \quad \bar{x}\in \Gamma , \, t\in [-\delta _0, \delta _0]. \end{aligned}$$

We denote the maps $\Gamma (\delta _0)\rightarrow \Gamma$; $x\mapsto \bar{x}$ and $\Gamma (\delta _0)\rightarrow \mathbb {R}$; $x\mapsto t$ by $\pi (x)$ and d(x), respectively (actually, $\pi$ is an orthogonal projection to $\Gamma$ and d agrees with the signed-distance function). The regularity of $\Omega$ is inherited to that of $\pi$, d, and n (cf. [7, Section 7.8]).

In [12, Section 8] we proved that $\pi |_{\Gamma _h}$ gives a homeomorphism (and piecewisely a diffeomorphism) between $\Gamma$ and $\Gamma _h$ provided h is sufficiently small, taking advantage of the fact that $\Gamma _h$ can be regarded as a linear interpolation of $\Gamma$ (recall the assumption on $\mathcal {S}_h$ mentioned above). If we write its inverse map $\pi ^*{:}\,\Gamma \rightarrow \Gamma _h$ as $\pi ^*(x) = \bar{x} + t^*(\bar{x}) n(\bar{x})$, then $t^*$ satisfies the estimates $\Vert \nabla _\Gamma ^k t^*\Vert _{L^\infty (\Gamma )} \le C_{kE}h^{2-k}$ for $k=0,1,2$, where $\nabla _\Gamma$ means the surface gradient along $\Gamma$ and where the constant depends on the $C^{1,1}$-regularity of $\Omega$. This in particular implies that $\Omega _h\triangle \Omega := (\Omega _h{\setminus }\Omega ) \cup (\Omega {\setminus }\Omega _h)$ and $\Gamma _h \cup \Gamma$ are contained in $\Gamma (\delta )$ with $\delta := C_{0E}h^2 < \delta _0$. We refer to $\Omega _h\triangle \Omega$, $\Gamma (\delta )$ and their subsets as boundary-skin layers or more simply as boundary skins.

Furthermore, we know from [12, Section 8] the following boundary-skin estimates:

$$\begin{aligned} \begin{aligned} \left| \int _\Gamma f\,d\gamma - \int _{\Gamma _h} f\circ \pi \,d\gamma _h \right|&\le C\delta \Vert f\Vert _{L^1(\Gamma )}, \\ \Vert f\Vert _{L^p(\Gamma (\delta ))}&\le C(\delta ^{1/p} \Vert f\Vert _{L^p(\Gamma )} + \delta \Vert \nabla f\Vert _{L^p(\Gamma (\delta ))}), \\ \Vert f - f\circ \pi \Vert _{L^p(\Gamma _h)}&\le C\delta ^{1-1/p} \Vert \nabla f\Vert _{L^p(\Gamma (\delta ))}, \end{aligned} \end{aligned}$$

(2.1)

where one can replace $\Vert f\Vert _{L^1(\Gamma )}$ in (2.1)$_1$ by $\Vert f\Vert _{L^1(\Gamma _h)}$. As a version of (2.1)$_2$, we also need

$$\begin{aligned} \Vert f\Vert _{L^p(\Omega _h{\setminus }\Omega )} \le C(\delta ^{1/p} \Vert f\Vert _{L^p(\Gamma _h)} + \delta \Vert \nabla f\Vert _{L^p(\Omega _h{\setminus }\Omega )}), \end{aligned}$$

(2.2)

whose proof will be given in Lemma . Finally, denoting by $n_h$ the outward unit normal to $\Gamma _h$, we notice that its error compared with n is estimated as $\Vert n\circ \pi - n_h\Vert _{L^\infty (\Gamma _h)} \le Ch$ (see [12, Section 9]).

2.3 Extension operators

We let $\tilde{\Omega }:= \Omega \cup \Gamma (\delta ) = \Omega _h\cup \Gamma (\delta )$ with $\delta = C_{0E}h^2$ given above. For $u \in W^{2,\infty }(\Omega ), f \in L^\infty (\Omega )$, and $\tau \in L^\infty (\Gamma )$, we assume that there exist extensions $\tilde{u} \in W^{2,\infty }(\tilde{\Omega }), \tilde{f} \in L^\infty (\tilde{\Omega })$, and $\tilde{\tau }\in L^\infty (\tilde{\Omega })$, respectively, which are stable in the sense that the norms of the extended quantities can be controlled by those of the original ones, e.g., $\Vert \tilde{u}\Vert _{W^{2,\infty }(\tilde{\Omega })} \le C\Vert u\Vert _{W^{2,\infty }(\Omega )}$. We emphasize that (1.1) would not hold any longer in the extended region $\tilde{\Omega }{\setminus } \overline{\Omega }$.

We also need extensions whose behavior in $\Gamma (\delta ){\setminus }\Omega$ can be completely described by that in $\Gamma (c\delta ) \cap \Omega$ for some constant $c>0$. To this end we introduce an extension operator $P{:}\,W^{k,p}(\Omega ) \rightarrow W^{k,p}(\tilde{\Omega }) \, (k=0,1,2, p \in [1,\infty ])$ as follows. For $x \in \Omega {\setminus }\Gamma (\delta )$ we let $Pf(x) = f(x)$; for $x = \bar{x} + tn(\bar{x}) \in \Gamma (\delta )$ we define

$$\begin{aligned} Pf(\bar{x} + tn(\bar{x})) = {\left\{ \begin{array}{ll} f(\bar{x} + t n(\bar{x})) &{}\quad (-\,\delta _0\le t<0), \\ 3f(\bar{x} - tn(\bar{x})) - 2f(\bar{x} - 2t n(\bar{x})) &{}\quad (0\le t\le \delta _0), \end{array}\right. } \quad \bar{x} \in \Gamma . \end{aligned}$$

Proposition 2.1

The extension operator P satisfies the following stability condition:

$$\begin{aligned} \Vert Pf\Vert _{W^{k, p}(\Gamma (\delta ))}&\le C\Vert f\Vert _{W^{k, p}(\Omega \cap \Gamma (2\delta ))} \quad (k=0,1,2), \quad p\in [1,\infty ], \end{aligned}$$

where C is independent of $\delta$ and f.

The proof of this proposition will be given in Theorem .

2.4 Dyadic decomposition

We introduce a dyadic decomposition of a domain according to [14]. Let $B(x_0; r) = \{x \in \mathbb {R}^N{:}\, |x - x_0| \le r\}$ and $A(x_0; r, R) = \{x \in \mathbb {R}^N{:}\, r\le |x - x_0|\le R\}$ denote a closed ball and annulus in $\mathbb {R}^N$ respectively.

Definition 2.1

For $x_0 \in \mathbb {R}^N, d_0>0, J \in \mathbb {N}_{\ge 0}$, the family of sets $\mathcal {A}(x_0, d_0, J) = \{A_j\}_{j=0}^J$ defined by

$$\begin{aligned} A_0 = B(x_0; d_0), \quad A_j = A(x_0; d_{j-1}, d_j), \quad d_j = 2^j d_0 \quad (j=1,\dots ,J) \end{aligned}$$

is called the dyadicJannuli with the center$x_0$and the initial stride$d_0$.

With a center and an initial stride specified, one can assign to a given domain a unique decomposition by dyadic annuli as follows.

Lemma 2.1

For a bounded domain $D \subset \mathbb {R}^N$, let $x_0 \in D$, $0<d_0<{\text {diam}}\,D$, and J be the smallest integer that is greater than $J' := \frac{\log ( {\text {diam}}\,D/d_0 )}{\log 2}$. Then we have $\overline{D} \subset \bigcup \mathcal {A}(x_0, d_0, J)$.

Proof

Since $2^{J'}d_0 = {\text {diam}}\,D$ and $J'<J\le J'+1$, one has ${\text {diam}}\,D<d_J\le 2{\text {diam}}D$. For arbitrary $x\in D$ we see that $|x - x_0| \le {\text {diam}}\,D<d_J$, which implies $\overline{D} \subset B(x_0; d_J) = \bigcup \mathcal {A}(x_0, d_0, J)$. $\square$

Definition 2.2

We define the decomposition ofDinto dyadic annuli with the center$x_0$and the initial stride$d_0$by$\mathcal {A}_D(x_0, d_0) = \{D\cap A_j\}_{j=0}^J$, where $\{A_j\}_{j=0}^J = \mathcal {A}(x_0, d_0, J)$ are the dyadic annuli given in Lemma . We also use the terminology dyadic decomposition for abbreviation.

For $\mathcal {A}(x_0, d_0, J) = \{A_j\}_{j=0}^J$ and $s \in [0,1]$, we consider expanded annuli $\mathcal {A}^{(s)}(x_0, d_0, J) = \{A_j^{(s)}\}_{j=0}^J$, where

$$\begin{aligned} A_0^{(s)}= & {} B(x_0; (1+s)d_0),\\ A_j^{(s)}:= & {} A(x_0; (1- \frac{s}{2})d_{j-1}, (1+s)d_j) \quad (j = 1, \dots , J). \end{aligned}$$

In particular, for $s=1$ one has $A_j^{(1)} = A_{j-1}\cup A_j\cup A_{j+1}$ where we set $A_{-1} := \emptyset$ and $A_{J+1} := A(x_0; d_J, d_{J+1})$ with $d_{J+1} := 2d_J$.

We collect some basic properties of weighted $L^p$-norms defined on a dyadic decomposition.

Lemma 2.2

For a dyadic decomposition $\mathcal {A}_D(x_0, d_0) = \{ D\cap A_j \}_{j=0}^J$ of D and $p\in [1,\infty ]$, the following estimates hold:

$$\begin{aligned} \Vert f\Vert _{L^1(D)}&\le \alpha _N^{1/p'} \sum _{j=0}^J d_j^{N/p'} \Vert f\Vert _{L^p(D\cap A_j)}, \end{aligned}$$

(2.3)

$$\begin{aligned} \sum _{j=0}^J d_j^{N/p'} \Vert f\Vert _{L^p(D\cap A_j^{(1)})}&\le (2^{N/p'} + 1 + 2^{-N/p'}) \sum _{j=0}^J d_j^{N/p'} \Vert f\Vert _{L^p(D\cap A_j)}. \end{aligned}$$

(2.4)

Here, $\alpha _N = \frac{2\pi ^{N/2}}{N\Gamma (N/2)}$ means the volume of the N-dimensional unit ball and $p' = p/(p-1)$.

Proof

It follows from the Hölder inequality that

$$\begin{aligned} \Vert f\Vert _{L^1(D)} = \sum _{j=0}^J \Vert f\Vert _{L^1(D\cap A_j)} \le \sum _{j=0}^J |A_j|^{1/p'} \Vert f\Vert _{L^p(D\cap A_j)}, \end{aligned}$$

which combined with $|A_j| = (1 - 2^{-N})d_j^N \alpha _N$ yields (2.3). The estimate (2.4) follows from the fact that

$$\begin{aligned} \Vert f\Vert _{L^p(D\cap A_j^{(1)})} \le \Vert f\Vert _{L^p(D\cap A_{j-1})} + \Vert f\Vert _{L^p(D\cap A_j)} + \Vert f\Vert _{L^p(D\cap A_{j+1})}, \end{aligned}$$

together with $D \cap A_{-1} = D \cap A_{J+1} = \emptyset$. $\square$

Setting now D to be $\Omega _h$ introduced in Sect. , we consider its dyadic decomposition $\mathcal {A}_{\Omega _h}(x_0, d_0) = \{\Omega _h \cap A_j\}_{j=0}^J$ and its triangulation $\mathcal {T}_h$. At this stage, each triangle in $\mathcal {T}_h$ can simultaneously intersect with some annuls A and its complement $A^c$; however, we have the following lemma:

Lemma 2.3

Let $\mathcal {A}_{\Omega _h}(x_0, d_0) = \{\Omega _h \cap A_j\}_{j=0}^J$ be a dyadic decomposition of $\Omega _h$ with $x_0 \in \Omega _h$ and $d_0 \in [16h, 1]$, and let $s\in [0, 3/4]$.

• (i) If $T\in \mathcal {T}_h$ satisfies $T \cap A_j^{(s)} \ne \emptyset$ then $M_T \subset A_j^{(s+1/4)}$, where $M_T$ is the macro element of T.

• (ii) If $T\in \mathcal {T}_h$ satisfies $T {\setminus } A_j^{(s+1/4)} \ne \emptyset$ then $M_T \subset (A_j^{(s)})^c$.

Proof

We only prove (i) since item (ii) can be shown similarly. Let $x \in M_T$ be arbitrary. By assumption there exists $x' \in T \cap A_j^{(s)}$; in particular, $(1-s/2)d_{j-1} \le |x' - x_0| \le (1+s)d_j$. Also, by definition of $M_T$, $|x - x'| \le 2h$. Then we have $(7/8 - s/2)d_{j-1} \le |x - x_0| \le (5/4 + s)d_j$ as a result of triangle inequalities, which implies $x \in A_j^{(s+1/4)}$. $\square$

Corollary 2.1

Under the assumption of Lemma , let $v \in H^1(\overline{\Omega }_h)$ satisfy ${\text {supp}}v \subset A_j^{(s)}$. Then we have ${\text {supp}}\mathcal {I}_hv \subset A_j^{(s+1/4)}$.

Proof

It suffices to show $\mathcal {I}_hv(x) = 0$ for all $x \in \Omega _h{\setminus } A_j^{(s+1/4)}$. In fact, since there exists $T \in \mathcal {T}_h$ such that $x \in T$, one has $M_T \cap A_j^{(s)} = \emptyset$ as a result of Lemma (ii). Hence $v|_{M_T} = 0$, so that $\mathcal {I}_hv|_T = 0$. $\square$

Finally, notice that for any dyadic decomposition $\mathcal {A}_{\Omega _h}(x_0, d_0)$ we have

$$\begin{aligned} \sum _{j=0}^J d_j^{\beta } \le {\left\{ \begin{array}{ll} Cd_J^\beta &{}\quad (\beta >0), \\ C(1 + |\log d_0|) &{}\quad (\beta = 0), \\ Cd_0^\beta &{}\quad (\beta <0), \end{array}\right. } \end{aligned}$$

(2.5)

where $C = C(N, \Omega , \beta )$ is independent of $x_0, d_0$, and J (for the case $\beta = 0$, recall Lemma to estimate J). Moreover, since $d_j \le d_J \le 2{\text {diam}}\, \Omega _h$, one has

$$\begin{aligned} d_j^\alpha + d_j^\beta \le Cd_j^{\min \{\alpha , \beta \}}, \quad 0\le j\le J, \; \alpha ,\beta \in \mathbb {R}, \; C=C(N, \Omega , \alpha , \beta ), \end{aligned}$$

which will not be emphasized in the subsequent arguments.

3 Main theorem and its reduction to $W^{1,1}$-analysis

Let us state the main result of this paper.

Theorem 3.1

Let $u\in W^{2,\infty }(\Omega )$ and $u_h \in V_h$ be the solutions of (1.1) and (1.2) respectively. Then there exists $h_0>0$ such that for all $h \in (0, h_0]$ and $v_h \in V_h$ we have

$$\begin{aligned} \Vert \tilde{u} - u_h\Vert _{L^\infty (\Omega _h)}&\le Ch|\log h| \, \Vert \tilde{u} - v_h\Vert _{W^{1,\infty }(\Omega _h)} + Ch^2|\log h| \, \Vert u\Vert _{W^{2,\infty }(\Omega )}, \\ \Vert \tilde{u} - u_h\Vert _{W^{1,\infty }(\Omega _h)}&\le C \Vert \tilde{u} - v_h\Vert _{W^{1,\infty }(\Omega _h)} + Ch \, \Vert u\Vert _{W^{2,\infty }(\Omega )}, \end{aligned}$$

where C is independent of h, u, and $v_h$.

Remark 3.1

• (i) By taking $v_h = \mathcal {I}_h \tilde{u}$, we immediately obtain (1.3).

• (ii) The factor $h \Vert \tilde{u} - v_h\Vert _{W^{1,\infty }(\Omega _h)}$ in the $L^\infty$-estimate could be replaced by $\Vert \tilde{u} - v_h\Vert _{L^\infty (\Omega _h)}$ (cf. [14, p. 889]), which will be discussed elsewhere.

• (iii) The above error estimates cannot be improved even if one employs a higher order finite element, as far as the boundary $\Gamma$ is only linearly approximated. In fact, the domain perturbation term $I_4$ (see Lemmas 3.2 and 3.5 below) gives rise to $O(h^2|\log h|)$- and O(h)-contributions for $L^\infty$- and $W^{1,\infty }$-error estimates respectively, regardless of the choice of $V_h$. Both of the approximation of functions and that of the boundary must be made higher order in a proper manner to achieve better accuracy (a typical way to do this is the use of isoparametric elements).

Let us reduce pointwise error estimates to $W^{1,1}$-error analysis for regularized Green’s functions, which is now a standard approach in this field. For arbitrary $T \in \mathcal {T}_h$ and $x_0 \in T$ we let $\eta = \eta _{x_0} \in C_0^\infty (T)$, $\eta \ge 0$ be a regularized delta function such that

$$\begin{aligned}&\int _T \eta (x) v_h(x) \, dx = v_h(x_0) \quad \forall v_h\in P_1(T), \quad \Vert \nabla ^k\eta \Vert _{L^\infty (T)} \le Ch^{-k} \, (k=0,1,2), \nonumber \\&\quad {\text {dist}}({\text {supp}}\eta , \partial T) \ge Ch, \end{aligned}$$

(3.1)

where C is independent of T, h, and $x_0$ (see [15] for construction of $\eta$).

Remark 3.2

• (i) The quasi-uniformity of meshes are needed to ensure the last two properties of (3.1).

• (ii) We have ${\text {supp}}\eta \cap \Gamma (2\delta ) = \emptyset$ with $\delta = C_{0E}h^2$, provided that h is sufficiently small.

We consider two kinds of regularized Green’s functions $g_0, g_1 \in C^\infty (\overline{\Omega })$ satisfying the following PDEs:

$$\begin{aligned} -\,\Delta g_0 + g_0 = \eta \quad \text {in}\quad \Omega , \quad \partial _ng_0 = 0 \quad \text {on}\quad \Gamma , \end{aligned}$$

and

$$\begin{aligned} -\,\Delta g_1 + g_1 = \partial \eta \quad \text {in}\quad \Omega , \quad \partial _ng_1 = 0 \quad \text {on}\quad \Gamma , \end{aligned}$$

where $\partial$ stands for an arbitrary directional derivative. Accordingly, we let $g_{0h}, g_{1h} \in V_h$ be the solutions for finite element approximate problems as follows:

$$\begin{aligned} a_h(v_h, g_{0h}) = (v_h, \eta )_{\Omega _h} \quad \forall v_h\in V_h, \quad \text {and}\quad a_h(v_h, g_{1h}) = (v_h, \partial \eta )_{\Omega _h} \quad \forall v_h\in V_h. \end{aligned}$$

The goal of this section is then to reduce Theorem to the estimate

$$\begin{aligned} \Vert \tilde{g}_m - g_{mh}\Vert _{W^{1,1}(\Omega _h)} \le C(h|\log h|)^{1-m}, \quad m=0,1, \end{aligned}$$

(3.2)

where C is independent of $h, x_0$, and $\partial$, and $\tilde{g}_m := Pg_m$ means the extension defined in Sect. . To observe this fact, we represent pointwise errors at $x_0$, with the help of $\eta$, as

$$\begin{aligned} \tilde{u}(x_0) - u_h(x_0)&= (\tilde{u} - v_h)(x_0) + (v_h - \tilde{u}, \eta )_{\Omega _h} + (\tilde{u} - u_h, \eta )_{\Omega _h}, \\ \partial (\tilde{u} - u_h)(x_0)&= \partial (\tilde{u} - v_h)(x_0) + (\partial (v_h - \tilde{u}), \eta )_{\Omega _h} - (\tilde{u} - u_h, \partial \eta )_{\Omega _h}, \end{aligned}$$

for all $v_h \in V_h$. Since the first two terms on the right-hand sides are bounded by $2\Vert \tilde{u} - v_h\Vert _{L^\infty (\Omega _h)}$ and $2\Vert \nabla (\tilde{u} - v_h)\Vert _{L^\infty (\Omega _h)}$, in order to prove Theorem it suffices to show that

$$\begin{aligned} |(\tilde{u} - u_h, \eta )_{\Omega _h}|&\le C h|\log h| \Vert \tilde{u} - v_h\Vert _{W^{1,\infty }(\Omega _h)} + Ch^2 |\log h| \Vert u\Vert _{W^{2,\infty }(\Omega )}, \\ |(\tilde{u} - u_h, \partial \eta )_{\Omega _h}|&\le C\Vert \tilde{u} - v_h\Vert _{W^{1,\infty }(\Omega _h)} + Ch \Vert u\Vert _{W^{2,\infty }(\Omega )}. \end{aligned}$$

With this aim we prove:

Proposition 3.1

For $m=0,1$ and arbitrary $v_h \in V_h$, one obtains

$$\begin{aligned} |(\tilde{u} - u_h, \partial ^m\eta )_{\Omega _h}|&\le C(\Vert \tilde{u} - v_h\Vert _{W^{1,\infty }(\Omega _h)} + Ch \Vert u\Vert _{W^{2,\infty }(\Omega )}) \Vert \tilde{g}_m - g_{mh}\Vert _{W^{1,1}(\Omega _h)} \\&\quad +\,Ch (h|\log h|)^{1-m} \Vert u\Vert _{W^{2,\infty }(\Omega )}. \end{aligned}$$

It is immediate to conclude Theorem from Proposition combined with (3.2). The rest of this section is thus devoted to the proof of Proposition , whereas (3.2) will be established in Sects. 4–6 below. From now on, we suppress the subscript m of $g_m$ and $g_{mh}$ for simplicity, as far as there is no fear of confusion.

Let us proceed to the proof of Proposition . Define functionals for $v \in H^1(\Omega _h)$, which will represent “residuals” of Galerkin orthogonality relation, by

$$\begin{aligned} {\text {Res}}_u(v)&= (-\,\Delta \tilde{u} + \tilde{u} - \tilde{f}, v)_{\Omega _h {\setminus } \Omega } + (\partial _{n_h} \tilde{u} - \tilde{\tau }, v)_{\Gamma _h}, \\ {\text {Res}}_g(v)&= (v, -\,\Delta \tilde{g} + \tilde{g})_{\Omega _h {\setminus } \Omega } + (v, \partial _{n_h} \tilde{g})_{\Gamma _h}. \end{aligned}$$

If in addition $v \in H^1(\tilde{\Omega })$ in the expanded domain $\tilde{\Omega }= \Omega \cup \Gamma (\delta )$, then ${\text {Res}}_u(v)$ admits another expression. To observe this, we introduce “signed” integration defined as follows:

$$\begin{aligned} (\phi , \psi )_{\Omega _h \triangle \Omega }'&:= (\phi , \psi )_{\Omega _h {\setminus } \Omega } - (\phi , \psi )_{\Omega {\setminus } \Omega _h}, \\ (\phi , \psi )_{\Gamma _h \cup \Gamma }'&:= (\phi , \psi )_{\Gamma _h} - (\phi , \psi )_{\Gamma }, \\ a_{\Omega _h \triangle \Omega }'(\phi , \psi )&:= (\nabla \phi , \nabla \psi )_{\Omega _h \triangle \Omega }' + (\phi , \psi )_{\Omega _h \triangle \Omega }'. \end{aligned}$$

Lemma 3.1

For $v \in H^1(\tilde{\Omega })$ we have

$$\begin{aligned} {\text {Res}}_u(v) = -(\tilde{f}, v)_{\Omega _h \triangle \Omega }' - (\tilde{\tau }, v)_{\Gamma _h \cup \Gamma }' + a_{\Omega _h \triangle \Omega }'(\tilde{u}, v). \end{aligned}$$

Proof

Notice that the following integration by parts formula holds:

$$\begin{aligned} (-\,\Delta \tilde{u}, v)_{\Omega _h \triangle \Omega }' = (\nabla \tilde{u}, \nabla v)_{\Omega _h \triangle \Omega }' - (\partial _{n_h}\tilde{u}, v)_{\Gamma _h} + (\partial _{n} u, v)_{\Gamma }. \end{aligned}$$

From this formula and (1.1) it follows that

$$\begin{aligned} (-\,\Delta \tilde{u}, v)_{\Omega _h {\setminus } \Omega } + (\partial _{n_h} \tilde{u}, v)_{\Gamma _h}= & {} (-\,\Delta u, v)_{\Omega {\setminus } \Omega _h} + (\nabla \tilde{u}, \nabla v)_{\Omega _h \triangle \Omega }' + (\partial _{n} u, v)_{\Gamma } \\= & {} -(u - f, v)_{\Omega {\setminus } \Omega _h} + (\nabla \tilde{u}, \nabla v)_{\Omega _h \triangle \Omega }' + (\tau , v)_{\Gamma }. \end{aligned}$$

Substituting this into the definition of ${\text {Res}}_u(v)$ leads to the desired equality. $\square$

Now we show that ${\text {Res}}_u(\cdot )$ and ${\text {Res}}_g(\cdot )$ represent residuals of Galerkin orthogonality relation for $\tilde{u} - u_h$ and $\tilde{g} - g_h$, respectively.

Lemma 3.2

For all $v_h \in V_h$ we have

$$\begin{aligned} a_h(\tilde{u} - u_h, v_h) = {\text {Res}}_u(v_h), \quad a_h(v_h, \tilde{g} - g_h) = {\text {Res}}_g(v_h), \end{aligned}$$

and

$$\begin{aligned} (\tilde{u} - u_h, \partial ^{m}\eta )_{\Omega _h}&= a_h(\tilde{u} - v_h, \tilde{g} - g_h) - {\text {Res}}_g(\tilde{u} - v_h) - {\text {Res}}_u(\tilde{g} - g_h) + {\text {Res}}_u(\tilde{g}) \\&=: I_1 + I_2 + I_3 + I_4. \end{aligned}$$

Proof

From integration by parts and from the definitions of u and $u_h$ we have

$$\begin{aligned} a_h(\tilde{u} - u_h, v_h)&= (-\,\Delta \tilde{u} + \tilde{u}, v_h)_{\Omega _h} + (\partial _{n_h}\tilde{u}, v_h)_{\Gamma _h} - (\tilde{f}, v_h)_{\Omega _h} - (\tilde{\tau }, v_h)_{\Gamma _h}\\&= {\text {Res}}_u(v_h). \end{aligned}$$

The second equality is obtained in the same way. To show the third equality, we observe that

$$\begin{aligned} (v_h - u_h, \partial ^m\eta )_{\Omega _h}&= a_h(v_h - u_h, g_h) = a_h(v_h - \tilde{u}, g_h) + a_h(\tilde{u} - u_h, g_h) \\&= a_h(\tilde{u} - v_h, \tilde{g} - g_h) - a_h(\tilde{u} - v_h, \tilde{g}) + {\text {Res}}_u(g_h). \end{aligned}$$

It follows from integration by parts, $-\,\Delta g + g = \partial ^m\eta$ in $\Omega$, and ${\text {supp}}\eta \subset \Omega _h\cap \Omega$, that

$$\begin{aligned} a_h(\tilde{u} - v_h, \tilde{g})&= (\tilde{u} - v_h, -\,\Delta \tilde{g} + \tilde{g})_{\Omega _h} + (\tilde{u} - v_h, \partial _{n_h}\tilde{g})_{\Gamma _h}\\&= (u - v_h, \partial ^m\eta )_{\Omega _h\cap \Omega } + {\text {Res}}_g(\tilde{u} - v_h) \\&= (\tilde{u} - v_h, \partial ^m\eta )_{\Omega _h} + {\text {Res}}_g(\tilde{u} - v_h). \end{aligned}$$

Combining the two relations above yields the third equality. $\square$

By the Hölder inequality, $|I_1| \le \Vert \tilde{u} - v_h\Vert _{W^{1,\infty }(\Omega _h)} \Vert \tilde{g} - g_h\Vert _{W^{1,1}(\Omega _h)}$. The other terms are estimated in the following three lemmas. There, boundary-skin estimates for g will be frequently exploited, which are collected in the appendix.

Lemma 3.3

$|I_2| \le C(h|\log h|)^{1-m} \, \Vert \tilde{u} - v_h\Vert _{L^\infty (\Omega _h)}$.

Proof

By the Hölder inequality,

$$\begin{aligned} |{\text {Res}}_g(\tilde{u} - v_h)| \le \Vert \tilde{u} - v_h\Vert _{L^\infty (\Omega _h)} (\Vert \tilde{g}\Vert _{W^{2,1}(\Gamma (\delta ))} + \Vert \partial _{n_h}\tilde{g}\Vert _{L^1(\Gamma _h)}), \end{aligned}$$

where $\Vert \tilde{g}\Vert _{W^{2,1}(\Gamma (\delta ))} \le Ch^{1-m}$ as a result of Corollary . Since $(\nabla g)\circ \pi \cdot n\circ \pi = 0$ on $\Gamma _h$, it follows again from Corollary that

$$\begin{aligned} \Vert \partial _{n_h}\tilde{g}\Vert _{L^1(\Gamma _h)}&\le \Vert \nabla \tilde{g} \cdot (n_h - n\circ \pi )\Vert _{L^1(\Gamma _h)} + \Vert \big ( \nabla \tilde{g} - (\nabla \tilde{g})\circ \pi \big ) \cdot n\circ \pi \Vert _{L^1(\Gamma _h)} \\&\le Ch \Vert \nabla \tilde{g}\Vert _{L^1(\Gamma _h)} + C\Vert \nabla ^2\tilde{g}\Vert _{L^1(\Gamma (\delta ))} \le C (h|\log h|)^{1-m} + Ch^{1-m}, \end{aligned}$$

which completes the proof. $\square$

Lemma 3.4

$|I_3| \le Ch\Vert u\Vert _{W^{2,\infty }(\Omega )} \Vert \tilde{g} - g_h\Vert _{W^{1,1}(\Omega _h)}$.

Proof

By the Hölder inequality and stability of extensions,

$$\begin{aligned} |{\text {Res}}_u(\tilde{g} - g_h)| \le C\Vert u\Vert _{W^{2, \infty }(\Omega )} \Vert \tilde{g} - g_h\Vert _{L^1(\Omega _h{\setminus }\Omega )} + \Vert \partial _{n_h}\tilde{u} - \tilde{\tau }\Vert _{L^\infty (\Gamma _h)} \Vert \tilde{g} - g_h\Vert _{L^1(\Gamma _h)}. \end{aligned}$$

From (2.2) and the trace theorem one has

$$\begin{aligned} \Vert \tilde{g} - g_h\Vert _{L^1(\Omega _h{\setminus }\Omega )}\le & {} C\delta (\Vert \tilde{g} - g_h\Vert _{L^1(\Gamma _h)} + \Vert \nabla (\tilde{g} - g_h)\Vert _{L^1(\Omega _h{\setminus }\Omega )})\\\le & {} Ch^2 \Vert \tilde{g} - g_h\Vert _{W^{1,1}(\Omega _h)}. \end{aligned}$$

From $(\nabla u)\circ \pi \cdot n\circ \pi = \tau \circ \pi$ on $\Gamma _h$, (2.1), and the stability of extensions, it follows that

$$\begin{aligned} \Vert \partial _{n_h}\tilde{u} - \tilde{\tau }\Vert _{L^\infty (\Gamma _h)}&\le \Vert \nabla \tilde{u}\cdot (n_h - n\circ \pi )\Vert _{L^\infty (\Gamma _h)} + \Vert \big ( \nabla \tilde{u} - (\nabla \tilde{u})\circ \pi \big ) \cdot n\circ \pi \Vert _{L^\infty (\Gamma _h)}\\&\quad +\,\Vert \tau \circ \pi - \tilde{\tau }\Vert _{L^\infty (\Gamma _h)} \\&\le Ch \Vert \nabla \tilde{u}\Vert _{L^\infty (\Gamma _h)} + C\delta \Vert \nabla ^2\tilde{u}\Vert _{L^\infty (\Gamma (\delta ))}\\&\quad +\,C\delta \Vert \nabla \tilde{\tau }\Vert _{L^\infty (\Gamma (\delta ))} \le Ch \Vert u\Vert _{W^{2,\infty }(\Omega _h)}. \end{aligned}$$

Combining the estimates above and using the trace theorem once again, we conclude

$$\begin{aligned} |{\text {Res}}_u(\tilde{g} - g_h)|&\le Ch^2 \Vert u\Vert _{W^{2,\infty }(\Omega )} \Vert \tilde{g} - g_h\Vert _{W^{1,1}(\Omega _h)} + Ch\Vert u\Vert _{W^{2,\infty }(\Omega _h)} \Vert \tilde{g} - g_h\Vert _{L^1(\Gamma _h)} \\&\le Ch\Vert u\Vert _{W^{2,\infty }(\Omega )} \Vert \tilde{g} - g_h\Vert _{W^{1,1}(\Omega _h)}. \end{aligned}$$

This completes the proof. $\square$

Lemma 3.5

$|I_4| \le Ch (h|\log h|)^{1-m} \Vert u\Vert _{W^{2,\infty }(\Omega )}$.

Proof

We recall from Lemma that

$$\begin{aligned} {\text {Res}}_u(\tilde{g}) = -(\tilde{f}, \tilde{g})_{\Omega _h \triangle \Omega }' - (\tilde{\tau }, \tilde{g})_{\Gamma _h \cup \Gamma }' + a_{\Omega _h \triangle \Omega }'(\tilde{u}, \tilde{g}). \end{aligned}$$

Let us estimate each term in the right-hand side. By (2.1)$_2$ we obtain

$$\begin{aligned} |(\tilde{f}, \tilde{g})_{\Omega _h \triangle \Omega }'| \le \Vert \tilde{f}\Vert _{L^\infty (\Gamma (\delta ))} \Vert \tilde{g}\Vert _{L^1(\Gamma (\delta ))} \le C\delta |\log h|^{1-m} \Vert u\Vert _{W^{2,\infty }(\Omega )}, \end{aligned}$$

where $\delta = C_{0E}h^2$. Next, from (2.1) and Corollary we find that

$$\begin{aligned} (\tilde{\tau }, \tilde{g})_{\Gamma _h \cup \Gamma }'&= |(\tau , g)_{\Gamma } - (\tilde{\tau }, \tilde{g})_{\Gamma _h}| \le |(\tau , g)_\Gamma - (\tau \circ \pi , g\circ \pi )_{\Gamma _h}| \\&\quad +\,|(\tau \circ \pi , g\circ \pi - \tilde{g})_{\Gamma _h}|+|(\tau \circ \pi - \tilde{\tau }, \tilde{g})_{\Gamma _h}| \\&\le C\delta \Vert \tau \Vert _{L^\infty (\Gamma )} \Vert g\Vert _{L^1(\Gamma )} + C\Vert \tau \Vert _{L^\infty (\Gamma )} \Vert \nabla \tilde{g}\Vert _{L^1(\Gamma (\delta ))}\\&\quad +\,C\delta \Vert \nabla \tilde{\tau }\Vert _{L^\infty (\Gamma (\delta ))} \Vert \tilde{g}\Vert _{L^1(\Gamma _h)} \\&\le C\delta \Vert \nabla u\Vert _{L^\infty (\Omega )} |\log h|^m + C\Vert \nabla u\Vert _{L^\infty (\Omega )} \delta h^{-m} |\log h|^{1-m}\\&\quad +\,C\delta \Vert u\Vert _{W^{2,\infty }(\Omega )} |\log h|^{1-m} \\&\le C\delta h^{-m} |\log h|^{1-m} \Vert u\Vert _{W^{2,\infty }(\Omega )}. \end{aligned}$$

Finally, for the last term we obtain

$$\begin{aligned} |a_{\Omega _h \triangle \Omega }'(\tilde{u}, \tilde{g})| \le \Vert \tilde{u}\Vert _{W^{1,\infty }(\Gamma (\delta ))} \Vert \tilde{g}\Vert _{W^{1,1}(\Gamma (\delta ))} \le C\Vert u\Vert _{W^{1,\infty }(\Omega )} \delta h^{-m} |\log h|^{1-m}. \end{aligned}$$

Collecting the above estimates proves the lemma. $\square$

Proposition is now an immediate consequence of Lemmas 3.2–3.5.

4 Weighted $H^1$-estimates

As a consequence of the previous section, we need to estimate $\Vert \tilde{g} - g_h\Vert _{W^{1,1}(\Omega _h)}$, where we keep dropping the subscript m (either 0 or 1) of $g_m$ and $g_{mh}$. To this end we introduce a dyadic decomposition $\mathcal {A}_{\Omega _h}(x_0, d_0) = \{\Omega _h\cap A_j\}_{j=0}^J$ of $\Omega _h$, and observe from (2.3) that

$$\begin{aligned} \Vert \tilde{g} - g_h\Vert _{W^{1,1}(\Omega _h)} \le C \sum _{j=0}^J d_j^{N/2} \Vert \tilde{g} - g_h\Vert _{H^1(\Omega _h \cap A_j)}. \end{aligned}$$

(4.1)

Then the weighted $H^1$-norm in the right-hand side is bounded as follows:

Proposition 4.1

There exists $K_0>0$ such that, for any dyadic decomposition $\mathcal {A}_{\Omega _h}(x_0, d_0) = \{\Omega _h \cap A_j\}_{j=0}^J$ of $\Omega _h$ with $d_0 = Kh$, $K \ge K_0$, we obtain

$$\begin{aligned} \sum _{j=0}^J d_j^{N/2} \Vert \tilde{g} - g_h\Vert _{H^1(\Omega _h \cap A_j)}\le & {} CK^{m+N/2} h^{1-m} + C(h|\log h|)^{1-m}\nonumber \\&\quad +\,C\sum _{j=0}^J d_j^{-1+N/2} \Vert \tilde{g} - g_h\Vert _{L^2(\Omega _h\cap A_j)}. \end{aligned}$$

(4.2)

Here the constants $K_0$ and C are independent of $h, x_0, \partial$, and K.

The rest of this section is devoted to the proof of the proposition above. In order to estimate $\Vert \tilde{g} - g_h\Vert _{H^1(\Omega _h \cap A_j)}$ for $j = 0, \dots , J$, we use a cut off function $\omega _j \in C_0^\infty (\mathbb {R}^N), \; \omega _j\ge 0$ such that

$$\begin{aligned} \omega _j \equiv 1 \quad \text {in}\quad A_j, \quad {\text {supp}}\omega _j \subset A_j^{(1/4)}, \quad \Vert \nabla ^k \omega _j\Vert _{L^\infty (\mathbb {R}^N)} \le Cd_j^{-k} \quad (k=0,1,2). \end{aligned}$$

(4.3)

Then we find that

$$\begin{aligned} \Vert \tilde{g} - g_h\Vert _{H^1(\Omega _h \cap A_j)}^2&\le \big ( \omega _j (\tilde{g} - g_h), \tilde{g} - g_h \big )_{\Omega _h} + \big ( \omega _j \nabla (\tilde{g} - g_h), \nabla (\tilde{g} - g_h) \big )_{\Omega _h} \\&= a_h\big ( \omega _j(\tilde{g} - g_h), \tilde{g} - g_h \big ) - \big ( \nabla \omega _j(\tilde{g} - g_h), \nabla (\tilde{g} - g_h) \big )_{\Omega _h} \\&= a_h\big ( \omega _j(\tilde{g} - g_h) - v_h, \tilde{g} - g_h \big )\\&\qquad -\,\big ( (\nabla \omega _j) (\tilde{g} - g_h), \nabla (\tilde{g} - g_h) \big )_{\Omega _h} + {\text {Res}}_g(v_h) \\&=: I_1 + I_2 + I_3, \end{aligned}$$

where $v_h \in V_h$ is arbitrary and we have used Lemma .

Substituting $v_h = \mathcal {I}_h(\omega _j(\tilde{g} - g_h))$, where $\mathcal {I}_h$ is the interpolation operator given in Sect. , we estimate $I_1, I_2$, and $I_3$ in the following.

Lemma 4.1

$I_1$ is bounded as

$$\begin{aligned} |I_1|&\le Chd_j^{-2} \Vert \tilde{g} - g_h\Vert _{L^2(\Omega _h \cap A_j^{(1/2)})} \Vert \tilde{g} - g_h\Vert _{H^1(\Omega _h \cap A_j^{(1/2)})}\nonumber \\&\quad + Chd_j^{-1} \Vert \tilde{g} - g_h\Vert _{H^1(\Omega _h \cap A_j^{(1/2)})}^2 \nonumber \\&\quad +\,C_j h d_j^{-m-N/2} \Vert \tilde{g} - g_h\Vert _{H^1(\Omega _h \cap A_j^{(1/2)})}, \end{aligned}$$

(4.4)

where $C_0 = C K^{m+N/2}$ and $C_j = C$ for $1\le j\le J$.

Proof

By Corollary we have ${\text {supp}}v_h \subset \Omega _h \cap A_j^{(1/2)}$, and hence

$$\begin{aligned} |I_1| \le \Vert \omega _j(\tilde{g} - g_h) - v_h\Vert _{H^1(\Omega _h)} \Vert \tilde{g} - g_h\Vert _{H^1(\Omega _h \cap A_j^{(1/2)})}. \end{aligned}$$

It follows from the interpolation error estimate, together with (4.3), that

$$\begin{aligned}&\Vert \omega _j(\tilde{g} - g_h) - v_h\Vert _{H^1(\Omega _h)}^2 \le Ch^2 \sum _{T \in \mathcal {T}_h} \Vert \nabla ^2 \big ( \omega _j(\tilde{g} - g_h) \big )\Vert _{L^2(T)}^2 \\&\quad \le Ch^2 \sum _{T\in \mathcal {T}_h} \Big ( \Vert (\nabla ^2\omega _j)(\tilde{g} - g_h)\Vert _{L^2(T)}^2 + \Vert (\nabla \omega _j) \otimes \nabla (\tilde{g} - g_h)\Vert _{L^2(T)}^2 + \Vert \nabla ^2\tilde{g}\Vert _{L^2(T)}^2 \Big ) \\&\quad \le Ch^2 \sum _{T \cap A_j^{(1/4)} \ne \emptyset } \Big ( d_j^{-4} \Vert \tilde{g} - g_h\Vert _{L^2(T)}^2 + d_j^{-2} \Vert \tilde{g} - g_h\Vert _{L^2(T)}^2 + \Vert \nabla ^2\tilde{g}\Vert _{L^2(T)}^2 \Big ), \end{aligned}$$

where we made use of the fact that $\nabla ^2g_h|_{T} \equiv 0$ for $T \in \mathcal {T}_h$. This combined with Lemma (i) implies

$$\begin{aligned} \Vert \omega _j(\tilde{g} - g_h) - v_h\Vert _{H^1(\Omega _h)}\le & {} Ch (d_j^{-2} \Vert \tilde{g} - g_h\Vert _{L^2(\Omega _h \cap A_j^{(1/2)})}\\&+\, d_j^{-1} \Vert \tilde{g} - g_h\Vert _{L^2(\Omega _h \cap A_j^{(1/2)})}\\&+\,\Vert \nabla ^2\tilde{g}\Vert _{L^2(\Omega _h \cap A_j^{(1/2)})}). \end{aligned}$$

When $j=0$, by the stability of extension and the $H^2$-regularity theory, we deduce that

$$\begin{aligned} \Vert \nabla ^2\tilde{g}\Vert _{L^2(\Omega _h\cap A_0^{(1/2)})} \le C\Vert g\Vert _{H^2(\Omega )} \le C\Vert \partial ^m\eta \Vert _{L^2(\Omega )} \le Ch^{-m-N/2} = CK^{m+N/2} d_0^{-m-N/2}. \end{aligned}$$

When $j \ge 1$, it follows from Lemma that $\Vert \nabla ^2\tilde{g}\Vert _{L^2(\Omega _h\cap A_0^{(1/2)})} \le Cd_j^{-m-N/2}$. Collecting the estimates above, we conclude (4.4). $\square$

For $I_2$ we have

$$\begin{aligned} |I_2| \le Cd_j^{-1} \Vert \tilde{g} - g_h\Vert _{L^2(\Omega _h \cap A_j^{(1/2)})} \Vert \tilde{g} - g_h\Vert _{H^1(\Omega _h \cap A_j^{(1/2)})}, \end{aligned}$$

which dominates the first term in the right-hand side of (4.4) because $hd_j^{-1} \le 1$.

Lemma 4.2

$|I_3| \le Ch d_j^{1/2-m-N/2} (\Vert \tilde{g} - g_h\Vert _{H^1(\Omega _h \cap A_j^{(1/4)})} + d_j^{-1} \Vert \tilde{g} - g_h\Vert _{L^2(\Omega _h \cap A_j^{(1/4)})})$.

Proof

Since $I_3 = (v_h, -\,\Delta \tilde{g} + \tilde{g})_{\Omega _h{\setminus }\Omega } + (v_h, \partial _{n_h}\tilde{g})_{\Gamma _h}$, we observe that

$$\begin{aligned} |(v_h, -\,\Delta \tilde{g} + \tilde{g})_{\Omega _h{\setminus }\Omega }|&\le C\delta ^{1/2} \Vert v_h\Vert _{H^1(\Omega _h)} (\delta d_j^{N-1})^{1/2} d_j^{-m-N}\\&\le C\delta d_j^{-1/2-m-N/2} \Vert v_h\Vert _{H^1(\Omega _h)}, \end{aligned}$$

and that

$$\begin{aligned} |(v_h, \partial _{n_h}\tilde{g})_{\Gamma _h}|&\le \Vert v_h\Vert _{L^2(\Gamma _h)} \Vert \partial _{n_h} \tilde{g}\Vert _{L^2(\Gamma _h \cap A_j^{(1/4)})} \\&\le C\Vert v_h\Vert _{H^1(\Omega _h)} \big ( \Vert \nabla \tilde{g} \cdot (n_h - n\circ \pi )\Vert _{L^2(\Gamma _h \cap A_j^{(1/4)})}\\&\quad +\,\Vert (\nabla \tilde{g} - (\nabla \tilde{g})\circ \pi ) \cdot n\circ \pi \Vert _{L^2(\Gamma _h \cap A_j^{(1/4)})} \big ) \\&\le C\Vert v_h\Vert _{H^1(\Omega _h)} \big ( h\Vert \nabla \tilde{g}\Vert _{L^2(\Gamma _h \cap A_j^{(1/4)})}\\&\quad + |\Gamma _h \cap A_j^{(1/4)}|^{1/2} \delta \Vert \nabla ^2\tilde{g}\Vert _{L^\infty (\Gamma _h \cap A_j^{(1/4)})} \big ) \\&\le C\Vert v_h\Vert _{H^1(\Omega _h)} (hd_j^{1/2 - m - N/2} + h^2d_j^{-1/2-m-N/2})\\&\le Ch d_j^{1/2-m-N/2} \Vert v_h\Vert _{H^1(\Omega _h)}. \end{aligned}$$

Therefore, by the $H^1$-stability of $\mathcal {I}_h$ and by $d_j \le 2{\text {diam}}\,\Omega$,

$$\begin{aligned} |I_3|&\le Ch d_j^{1/2-m-N/2} \Vert \omega _j(\tilde{g} - g_h)\Vert _{H^1(\Omega _h)} \\&\le Ch d_j^{1/2-m-N/2} (\Vert \tilde{g} - g_h\Vert _{H^1(\Omega _h \cap A_j^{(1/4)})} + d_j^{-1} \Vert \tilde{g} - g_h\Vert _{L^2(\Omega _h \cap A_j^{(1/4)})}), \end{aligned}$$

which completes the proof. $\square$

Collecting the estimates for $I_1, I_2$, and $I_3$ we deduce that

$$\begin{aligned}&d_j^{N/2} \Vert \tilde{g} - g_h\Vert _{H^1(\Omega _h \cap A_j)}\\&\quad \le C(hd_j^{-1})^{1/2} d_j^{N/2} \Vert \tilde{g} - g_h\Vert _{H^1(\Omega _h \cap A_j^{(1)})} \\&\qquad +\,C \left( d_j^{N/2}\Vert \tilde{g} - g_h\Vert _{H^1(\Omega _h \cap A_j^{(1)})}\right) ^{1/2} \left( d_j^{-1+N/2}\Vert \tilde{g} - g_h\Vert _{L^2(\Omega _h \cap A_j^{(1)})} \right) ^{1/2} \\&\qquad +\,\big ( C_j hd_j^{-m} (1 + d_j^{1/2}) \big )^{1/2} (d_j^{N/2}\Vert \tilde{g} - g_h\Vert _{H^1(\Omega _h \cap A_j^{(1)})})^{1/2} \\&\qquad +\,C (hd_j^{1/2-m})^{1/2} \left( d_j^{-1+N/2}\Vert \tilde{g} - g_h\Vert _{L^2(\Omega _h \cap A_j^{(1)})}\right) ^{1/2}. \end{aligned}$$

We now take the summation for $j = 0, 1, \dots , J$ and apply (2.4) to have

$$\begin{aligned} \sum _{j=0}^J d_j^{N/2} \Vert \tilde{g} - g_h\Vert _{H^1(\Omega _h \cap A_j)}&\le C'(hd_0^{-1})^{1/2} \sum _{j=0}^J d_j^{N/2} \Vert \tilde{g} - g_h\Vert _{H^1(\Omega _h\cap A_j)}\\&\quad +\,\frac{1}{4} \sum _{j=0}^J d_j^{N/2} \Vert \tilde{g} - g_h\Vert _{H^1(\Omega _h\cap A_j)} \\&\quad +\,\sum _{j=0}^J C_jh d_j^{-m}(1 + d_j^{1/2}) + Ch\sum _{j=0}^J d_j^{1/2-m}\\&\quad +\,C \sum _{j=0}^J d_j^{-1+N/2} \Vert \tilde{g} - g_h\Vert _{L^2(\Omega _h \cap A_j)}. \end{aligned}$$

If $hd_0^{-1} = K^{-1} \le 1/(4C')^2$, then one can absorb the first two terms into the left-hand side to conclude (4.2). This completes the proof of Proposition .

Thus we are left to deal with $\sum _{j=0}^J d_j^{-1+N/2} \Vert \tilde{g} - g_h\Vert _{L^2(\Omega _h \cap A_j)}$, which will be the scope of the next section.

5 Weighted $L^2$-estimates

Let us give estimation of the weighted $L^2$-norm appearing in the last term of (4.2).

Proposition 5.1

There exists $K_0>0$ such that, for any dyadic decomposition $\mathcal {A}_{\Omega _h}(x_0, d_0) = \{\Omega _h \cap A_j\}_{j=0}^J$ of $\Omega _h$ with $d_0 = Kh$, $K_0 \le K \le h^{-1}$, we obtain

$$\begin{aligned}&\sum _{j=0}^J d_j^{-1+N/2} \Vert \tilde{g} - g_h\Vert _{L^2(\Omega _h \cap A_j)} \nonumber \\&\quad \le C(hd_0^{-1}) \left( \sum _{j=0}^J d_j^{N/2} \Vert \tilde{g} - g_h\Vert _{H^1(\Omega _h\cap A_j)} + \Vert \tilde{g} - g_h\Vert _{W^{1,1}(\Omega _h)} \right) + Ch^{3/2 - m}, \end{aligned}$$

(5.1)

where the constants $K_0$ and C are independent of $h, x_0, \partial$, and K.

To prove this, first we fix $j=0, \dots , J$ and estimate $\Vert \tilde{g} - g_h\Vert _{L^2(\Omega _h \cap A_j)}$ based on a localized version of the Aubin–Nitsche trick. In fact, since

$$\begin{aligned} \Vert \tilde{g} - g_h\Vert _{L^2(\Omega _h \cap A_j)} = \sup _{\begin{array}{c} \varphi \in C_0^\infty (\Omega _h \cap A_j) \\ \Vert \varphi \Vert _{L^2(\Omega _h \cap A_j)} = 1 \end{array}} (\varphi , \tilde{g} - g_h)_{\Omega _h}, \end{aligned}$$

it suffices to examine $(\varphi , \tilde{g} - g_h)_{\Omega _h}$ for such $\varphi$. To express this quantity with a solution of a dual problem, we consider

$$\begin{aligned} -\,\Delta w + w = \varphi \quad \text {in}\quad \Omega , \quad \partial _n w = 0 \quad \text {on}\quad \Gamma , \end{aligned}$$

(5.2)

where $\varphi$ is extended by 0 to the outside of $\Omega _h \cap A_j$. From the elliptic regularity theory we know that the solution w is smooth enough. We then obtain the following:

Lemma 5.1

For all $w_h \in V_h$ we have

$$\begin{aligned} (\varphi , \tilde{g} - g_h)_{\Omega _h}&= a_h(\tilde{w} - w_h, \tilde{g} - g_h) - {\text {Res}}_w(\tilde{g} - g_h) - {\text {Res}}_g(\tilde{w} - w_h) + {\text {Res}}_g(\tilde{w}) \nonumber \\&=: I_1 + I_2 + I_3 + I_4, \end{aligned}$$

(5.3)

where $\tilde{w} := Pw$ and ${\text {Res}}_w : H^1(\Omega _h) \rightarrow \mathbb {R}$ is given by

$$\begin{aligned} {\text {Res}}_w(v) := (-\,\Delta \tilde{w} + \tilde{w} - \varphi , v)_{\Omega _h{\setminus }\Omega } + (\partial _{n_h}\tilde{w}, v)_{\Gamma _h}. \end{aligned}$$

Proof

We see that

$$\begin{aligned} (\varphi , \tilde{g} - g_h)_{\Omega _h}&= (\varphi , g - g_h)_{\Omega _h\cap \Omega } + (\varphi , \tilde{g} - g_h)_{\Omega _h{\setminus }\Omega } \\&= (-\,\Delta \tilde{w} + \tilde{w}, \tilde{g} - g_h)_{\Omega _h} + (\Delta \tilde{w} - \tilde{w} + \varphi , \tilde{g} - g_h)_{\Omega _h{\setminus }\Omega } \\&= a_h(\tilde{w}, \tilde{g} - g_h) - (\partial _{n_h}\tilde{w}, \tilde{g} - g_h)_{\Gamma _h} + (\Delta \tilde{w} - \tilde{w} + \varphi , \tilde{g} - g_h)_{\Omega _h{\setminus }\Omega } \\&= a_h(\tilde{w} - w_h, \tilde{g} - g_h) + {\text {Res}}_g(w_h) - {\text {Res}}_w(g - g_h), \end{aligned}$$

where we have used $a_h(w_h, \tilde{g} - g_h) = {\text {Res}}_g(w_h)$ from Lemma . This yields the desired equality. $\square$

Remark 5.1

In a similar way to Lemma , one can derive another expression for ${\text {Res}}_g(v)$ if $v \in H^1(\tilde{\Omega })$:

$$\begin{aligned} {\text {Res}}_g(v) = a_{\Omega _h\triangle \Omega }'(v, \tilde{g}). \end{aligned}$$

In the following four lemmas, taking $w_h = \mathcal {I}_h\tilde{w}$, we estimate $I_1, I_2, I_3$, and $I_4$ by dividing the integrals over $\Omega _h$, $\Gamma _h$, or boundary-skin layers, into those defined near $A_j$ and away from $A_j$. The former will be bounded, e.g., by the Hölder inequality of the form $\Vert \phi \Vert _{L^2(\Omega _h)} \Vert \psi \Vert _{L^2(\Omega _h \cap A_j^{(1/2)})}$ together with $H^2$-regularity estimates for w, whereas the latter will be bounded by $\Vert \phi \Vert _{L^\infty (\Omega _h {\setminus } A_j^{(1/2)})} \Vert \psi \Vert _{L^1(\Omega _h)}$ together with Green’s function estimates for w (see Lemma ).

Lemma 5.2

$|I_1| \le Ch \Vert \tilde{g} - g_h\Vert _{H^1(\Omega _h \cap A_j^{(1/2)})} + Chd_j^{-N/2} \Vert \tilde{g} - g_h\Vert _{W^{1,1}(\Omega _h)}.$

Proof

By the Hölder inequality mentioned above,

$$\begin{aligned} |I_1|&\le \Vert \tilde{w} - w_h\Vert _{H^1(\Omega _h)} \Vert \tilde{g} - g_h\Vert _{H^1(\Omega _h\cap A_j^{(1/2)})}\\&\quad + \Vert \tilde{w} - w_h\Vert _{W^{1,\infty }(\Omega _h{\setminus } A_j^{(1/2)})} \Vert \tilde{g} - g_h\Vert _{W^{1,1}(\Omega _h)}, \end{aligned}$$

where we notice that

$$\begin{aligned} \Vert \tilde{w} - w_h\Vert _{H^1(\Omega _h)} \le Ch \Vert w\Vert _{H^2(\Omega )} \le Ch \Vert \varphi \Vert _{L^2(\mathbb {R}^N)} = Ch, \end{aligned}$$

and from Lemma that

$$\begin{aligned} \Vert \tilde{w} - w_h\Vert _{W^{1,\infty }(\Omega _h{\setminus } A_j^{(1/2)})} \le Ch \Vert \nabla ^2\tilde{w}\Vert _{L^\infty (\Omega _h{\setminus } A_j^{(1/4)})} \le Chd_j^{-N/2}. \end{aligned}$$

This completes the proof. $\square$

Lemma 5.3

$I_2$ is bounded as

$$\begin{aligned}&|I_2| \le Ch^{1/2} \Vert \tilde{g} - g_h\Vert _{L^2(\Omega _h \cap A_j^{(3/4)})} + Ch \Vert \tilde{g} - g_h\Vert _{H^1(\Omega _h \cap A_j^{(3/4)})}\\&\quad +\, Chd_j^{-N/2} \Vert \tilde{g} - g_h\Vert _{W^{1,1}(\Omega _h)}. \end{aligned}$$

Proof

Recall that $I_2 = (\Delta \tilde{w} - \tilde{w} + \varphi , \tilde{g} - g_h)_{\Omega _h{\setminus }\Omega } - (\partial _{n_h}\tilde{w}, \tilde{g} - g_h)_{\Gamma _h} =: I_{21} + I_{22}$. Noting that $\varphi = 0$ in $\Omega _h{\setminus } A_j^{(1/2)}$ we estimate $I_{21}$ by

$$\begin{aligned} |I_{21}|&\le C\left( \Vert w\Vert _{H^2(\Omega )} + \Vert \varphi \Vert _{L^2(\mathbb {R}^N)}\right) \Vert \tilde{g} - g_h\Vert _{L^2\left( (\Omega _h{\setminus }\Omega ) \cap A_j^{(1/2)}\right) }\\&\quad +\,\Vert \tilde{w}\Vert _{W^{2,\infty }(\Omega _h{\setminus } A_j^{(1/2)})} \Vert \tilde{g} - g_h\Vert _{L^1(\Omega _h{\setminus }\Omega )} \\&\le C\Vert \tilde{g} - g_h\Vert _{L^2( (\Omega _h{\setminus }\Omega ) \cap A_j^{(1/2)})} + Cd_j^{-N/2} \Vert \tilde{g} - g_h\Vert _{L^1(\Omega _h{\setminus }\Omega )}. \end{aligned}$$

To address the first term we introduce $\omega _j' \in C_0^\infty (\mathbb {R}^N), \, \omega _j' \ge 0$ such that

$$\begin{aligned} \omega _j' \equiv 1 \quad \text {in}\quad A_j^{(1/2)}, \quad {\text {supp}}\omega _j' \subset A_j^{(3/4)}, \quad \Vert \nabla ^k \omega _j'\Vert _{L^\infty (\mathbb {R}^N)} \le Cd_j^{-k} \; (k=0,1,2). \end{aligned}$$

Then it follows from (2.2) and the trace estimate that

$$\begin{aligned}&\Vert \tilde{g} - g_h\Vert _{L^2( (\Omega _h{\setminus }\Omega ) \cap A_j^{(1/2)})}\nonumber \\&\quad \le \Vert \omega _j'(\tilde{g} - g_h)\Vert _{L^2(\Omega _h{\setminus }\Omega )} \nonumber \\&\quad \le C\delta ^{1/2} \Vert \omega _j' (\tilde{g} - g_h)\Vert _{L^2(\Gamma _h)} + C\delta \Vert \nabla \big ( \omega _j' (\tilde{g} - g_h) \big )\Vert _{L^2(\Omega _h{\setminus }\Omega )} \nonumber \\&\quad \le Ch \Vert \omega _j' (\tilde{g} - g_h)\Vert _{L^2(\Omega _h)}^{1/2} \Vert \omega _j' (\tilde{g} - g_h)\Vert _{H^1(\Omega _h)}^{1/2} \nonumber \\&\qquad +\,Ch^2d_j^{-1} \Vert \tilde{g} - g_h\Vert _{L^2(\Omega _h \cap A_j^{(3/4)})} + Ch^2 \Vert \nabla (\tilde{g} - g_h)\Vert _{L^2(\Omega _h \cap A_j^{(3/4)})} \nonumber \\&\quad \le Ch(1 + d_j^{-1/2}) \Vert \tilde{g} - g_h\Vert _{L^2(\Omega _h \cap A_j^{(3/4)})} + Ch \Vert \tilde{g} - g_h\Vert _{H^1(\Omega _h \cap A_j^{(3/4)})} \nonumber \\&\quad \le Ch^{1/2} \Vert \tilde{g} - g_h\Vert _{L^2(\Omega _h \cap A_j^{(3/4)})} + Ch \Vert \tilde{g} - g_h\Vert _{H^1(\Omega _h \cap A_j^{(3/4)})}, \end{aligned}$$

(5.4)

where we have used $hd_j^{-1} \le 1$ and $h\le 1$. Again by (2.2) we also have

$$\begin{aligned} \Vert \tilde{g} - g_h\Vert _{L^1(\Omega _h{\setminus }\Omega )}&\le C\delta (\Vert \tilde{g} - g_h\Vert _{L^1(\Gamma _h)} + \Vert \nabla (\tilde{g} - g_h)\Vert _{L^1(\Omega _h{\setminus }\Omega )})\\&\le Ch^2 \Vert \tilde{g} - g_h\Vert _{W^{1,1}(\Omega _h)}. \end{aligned}$$

Combining the estimates above now gives

$$\begin{aligned} |I_{21}| \le Ch^{1/2} \Vert \tilde{g} - g_h\Vert _{L^2(\Omega _h \cap A_j^{(3/4)})}+ & {} Ch \Vert \tilde{g} - g_h\Vert _{H^1(\Omega _h \cap A_j^{(3/4)})}\nonumber \\+ & {} Ch^2d_j^{-N/2} \Vert \tilde{g} - g_h\Vert _{W^{1,1}(\Omega _h)}. \end{aligned}$$

(5.5)

Next we estimate $I_{22}$ by

$$\begin{aligned} |I_{22}| \le \Vert \partial _{n_h}\tilde{w}\Vert _{L^2(\Gamma _h)} \Vert \tilde{g} - g_h\Vert _{L^2(\Gamma _h\cap A_j^{(1/2)})} + \Vert \partial _{n_h}\tilde{w}\Vert _{L^\infty (\Gamma _h {\setminus } A_j^{(1/2)})} \Vert \tilde{g} - g_h\Vert _{L^1(\Gamma _h)}. \end{aligned}$$

For the first term we see that

$$\begin{aligned} \Vert \partial _{n_h}\tilde{w}\Vert _{L^2(\Gamma _h)}&\le \Vert \nabla \tilde{w} \cdot (n_h - n\circ \pi )\Vert _{L^2(\Gamma _h)} + \Vert \big ( \nabla \tilde{w} - (\nabla \tilde{w})\circ \pi \big )\cdot n\circ \pi \Vert _{L^2(\Gamma _h)} \\&\le Ch \Vert \nabla \tilde{w}\Vert _{L^2(\Gamma _h)} + C\delta ^{1/2} \Vert \nabla ^2\tilde{w}\Vert _{L^2(\Gamma (\delta ))} \le Ch\Vert w\Vert _{H^2(\Omega )} \le Ch, \end{aligned}$$

and, in a similar way as we derived (5.4), that

$$\begin{aligned} \Vert \tilde{g} - g_h\Vert _{L^2(\Gamma _h\cap A_j^{(1/2)})} \le Cd_j^{-1/2} \Vert \tilde{g} - g_h\Vert _{L^2(\Omega _h \cap A_j^{(3/4)})} + C \Vert \tilde{g} - g_h\Vert _{H^1(\Omega _h \cap A_j^{(3/4)})}. \end{aligned}$$

For the second term, observe that

$$\begin{aligned} \Vert \partial _{n_h}\tilde{w}\Vert _{L^\infty (\Gamma _h {\setminus } A_j^{(1/2)})}&\le \Vert \nabla \tilde{w} \cdot (n_h - n\circ \pi )\Vert _{L^\infty (\Gamma _h {\setminus } A_j^{(1/2)})}\\&\quad +\,\Vert \big ( \nabla \tilde{w} - (\nabla \tilde{w})\circ \pi \big )\cdot n\circ \pi \Vert _{L^\infty (\Gamma _h {\setminus } A_j^{(1/2)})} \\&\le Ch \Vert \nabla \tilde{w}\Vert _{L^\infty (\Gamma (\delta ) {\setminus } A_j^{(1/2)} )} + C\delta \Vert \nabla ^2\tilde{w}\Vert _{L^\infty (\Gamma (\delta ) {\setminus } A_j^{(1/4)})} \\&\le Chd_j^{1-N/2} + Ch^2d_j^{-N/2} \le Chd_j^{1-N/2}, \end{aligned}$$

and that $\Vert \tilde{g} - g_h\Vert _{L^1(\Gamma _h)} \le C\Vert \tilde{g} - g_h\Vert _{W^{1,1}(\Omega _h)}$. Combining these estimates, we deduce

$$\begin{aligned} |I_{22}|\le & {} Chd_j^{-1/2} \Vert \tilde{g} - g_h\Vert _{L^2(\Omega _h \cap A_j^{(3/4)})} + Ch \Vert \tilde{g} - g_h\Vert _{H^1(\Omega _h \cap A_j^{(3/4)})}\nonumber \\&+ Chd_j^{1-N/2} \Vert \tilde{g} - g_h\Vert _{W^{1,1}(\Omega _h)}. \end{aligned}$$

(5.6)

From (5.5) and (5.6), together with $h\le d_j\le 2{\text {diam}}\,\Omega$, we conclude the desired estimate. $\square$

Lemma 5.4

$|I_3| \le Ch^{5/2-m} d_j^{-N/2}$.

Proof

Recall that $I_3 = (\tilde{w} - w_h, \Delta \tilde{g} - \tilde{g})_{\Omega _h{\setminus }\Omega } - (\tilde{w} - w_h, \partial _{n_h}\tilde{g})_{\Gamma _h} =: I_{31} + I_{32}$. We estimate $I_{31}$ by

$$\begin{aligned} |I_{31}|&\le \Vert \tilde{w} - w_h\Vert _{L^2(\Omega _h)} \Vert \tilde{g}\Vert _{H^2(\Gamma (\delta ) \cap A_j^{(1/2)})} + \Vert \tilde{w} - w_h\Vert _{L^\infty (\Omega _h {\setminus } A_j^{(1/2)})} \Vert \tilde{g}\Vert _{W^{2,1}(\Gamma (\delta ))} \\&\le Ch^2 \Vert \nabla ^2\tilde{w}\Vert _{L^2(\Omega _h)} (\delta d_j^{N-1})^{1/2} d_j^{-m-N} + Ch^2 \Vert \nabla ^2\tilde{w}\Vert _{L^\infty (\Omega _h {\setminus } A_j^{1/4})} \delta d_0^{-1-m} \\&\le Ch^3 d_j^{-1/2-m-N/2} + Ch^2d_j^{-N/2} h^{1-m} \le Ch^{5/2-m} d_j^{-N/2}, \end{aligned}$$

where we have used $h\le d_j$.

It remains to consider $I_{32}$; we estimate it by

$$\begin{aligned} \Vert \tilde{w} - w_h\Vert _{L^2(\Gamma _h)} \Vert \partial _{n_h}\tilde{g}\Vert _{L^2(\Gamma _h \cap A_j^{(1/2)})} + \Vert \tilde{w} - w_h\Vert _{L^\infty (\Gamma _h {\setminus } A_j^{(1/2)})} \Vert \partial _{n_h}\tilde{g}\Vert _{L^1(\Gamma _h)}. \end{aligned}$$

For the first term, we have $\Vert \tilde{w} - w_h\Vert _{L^2(\Gamma _h)} \le Ch^{3/2} \Vert \nabla ^2\tilde{w}\Vert _{L^2(\Omega _h)} \le Ch^{3/2}$ and

$$\begin{aligned}&\Vert \partial _{n_h}\tilde{g}\Vert _{L^2(\Gamma _h \cap A_j^{(1/2)})} \\&\quad \le |\Gamma _h \cap A_j^{(1/2)}|^{1/2} \big ( \Vert \nabla \tilde{g} \cdot (n_h - n\circ \pi )\Vert _{L^\infty (\Gamma _h \cap A_j^{(1/2)})}\nonumber \\&\qquad + \Vert \nabla \tilde{g} - (\nabla \tilde{g})\circ \pi \Vert _{L^\infty (\Gamma _h \cap A_j^{(1/2)})} \big ) \\&\quad \le Cd_j^{(N-1)/2} (h\Vert \nabla \tilde{g}\Vert _{L^\infty (\Gamma _h \cap A_j^{(1/2)})} + \delta \Vert \nabla ^2\tilde{g}\Vert _{L^\infty (\Gamma (\delta ) \cap A_j^{(3/4)})}) \\&\quad \le Cd_j^{(N-1)/2} (hd_j^{1-m-N} + h^2 d_j^{-m-N}) \le Chd_j^{1/2-m-N/2}. \end{aligned}$$

For the second term, we have $\Vert \tilde{w} - w_h\Vert _{L^\infty (\Gamma _h {\setminus } A_j^{(1/2)})} \le Ch^2\Vert \nabla ^2\tilde{w}\Vert _{L^\infty (\Gamma _h {\setminus } A_j^{(1/4)})} \le Ch^2d_j^{-N/2}$ and we find from Corollary that $\Vert \partial _{n_h}\tilde{g}\Vert _{L^1(\Gamma _h)} \le C(h |\log h|)^{1-m} \le Ch^{(1-m)/2}$. Therefore,

$$\begin{aligned} |I_{32}| \le C h^{5/2} d_j^{1/2-m-N/2} + Ch^{5/2-m/2} d_j^{-N/2} \le Ch^{5/2-m} d_j^{-N/2}, \end{aligned}$$

which completes the proof. $\square$

Lemma 5.5

$|I_4| \le Ch^2 d_j^{1/2-m-N/2} + Ch^{2-m} |\log h|^{1-m} d_j^{1-N/2}$.

Proof

We estimate $I_4 = a_{\Omega _h \triangle \Omega }'(\tilde{w}, \tilde{g})$ by

$$\begin{aligned} |I_4| \le \Vert \tilde{w}\Vert _{H^1(\Gamma (\delta ))} \Vert \tilde{g}\Vert _{H^1(\Gamma (\delta ) \cap A_j^{(1/2)})} + \Vert \tilde{w}\Vert _{W^{1,\infty }(\Gamma (\delta ) {\setminus } A_j^{(1/2)})} \Vert \tilde{g}\Vert _{W^{1,1}(\Gamma (\delta ))}. \end{aligned}$$

The first term of the right-hand side is bounded, using (2.1)$_2$ and Lemma , by

$$\begin{aligned} C\delta ^{1/2}\Vert w\Vert _{H^2(\Omega )} (\delta d_j^{N-1})^{1/2} d_j^{1-m-N} \le Ch^2 d_j^{1/2-m-N/2}. \end{aligned}$$

The second term is bounded, in view of Lemma and Corollary , by $Cd_j^{1-N/2} \delta h^{-m} |\log h|^{1-m}$. This completes the proof. $\square$

Now we substitute the results of Lemmas 5.2–5.5 into (5.3) and multiply by $d_j^{-1+N/2}$ to obtain

$$\begin{aligned}&d_j^{-1+N/2} \Vert \tilde{g} - g_h\Vert _{L^2(\Omega _h \cap A_j)} \nonumber \\&\quad \le C(hd_j^{-1}) d_j^{N/2} \Vert \tilde{g} - g_h\Vert _{H^1(\Omega _h \cap A_j^{(1)})} + C(hd_j^{-1}) \Vert \tilde{g} - g_h\Vert _{W^{1,1}(\Omega _h)} \nonumber \\&\quad +\,Ch^{1/2} d_j^{-1+N/2} \Vert \tilde{g} - g_h\Vert _{L^2(\Omega _h \cap A_j^{(1)})}\nonumber \\&\quad +\, Ch^{5/2-m} d_j^{-1} + Ch^2 d_j^{-1/2-m} + Ch^{2-m} |\log h|^{1-m}. \end{aligned}$$

(5.7)

Taking the summation for $j=0,\dots ,J$, assuming h is sufficiently small and using (2.4), we are able to absorb the third term in the right-hand side of (5.7) and then arrive at

$$\begin{aligned}&\sum _{j=0}^J d_j^{-1+N/2} \Vert \tilde{g} - g_h\Vert _{L^2(\Omega _h \cap A_j)}\\&\quad \le C(hd_0^{-1}) \left( \sum _{j=0}^J d_j^{N/2} \Vert \tilde{g} - g_h\Vert _{H^1(\Omega _h\cap A_j)} + \Vert \tilde{g} - g_h\Vert _{W^{1,1}(\Omega _h)} \right) \\&\qquad +\,Ch^{5/2-m} d_0^{-1} + Ch^2 d_0^{-1/2-m} + Ch^{2-m} |\log h|^{1-m} |\log d_0|, \end{aligned}$$

where we note that the last three terms can be estimated by $Ch^{3/2-m}$ because $d_0 = Kh\le 1$ and $K>1$. This completes the proof of Proposition .

6 End of the proof of the main theorem

Substituting (5.1) into (4.2) we obtain

$$\begin{aligned}&\sum _{j=0}^J d_j^{N/2} \Vert \tilde{g} - g_h\Vert _{H^1(\Omega _h \cap A_j)}\\&\quad \le C''K^{-1} \left( \sum _{j=0}^J d_j^{N/2} \Vert \tilde{g} - g_h\Vert _{H^1(\Omega _h\cap A_j)} + \Vert \tilde{g} - g_h\Vert _{W^{1,1}(\Omega _h)} \right) \\&\qquad +\,CK^{m+N/2}h^{1-m} + C(h|\log h|)^{1-m}. \end{aligned}$$

If $K \ge 2C''$, then it follows that

$$\begin{aligned}&\sum _{j=0}^J d_j^{N/2} \Vert \tilde{g} - g_h\Vert _{H^1(\Omega _h \cap A_j)}\\&\quad \le CK^{-1} \Vert \tilde{g} - g_h\Vert _{W^{1,1}(\Omega _h)} + CK^{m+N/2}h^{1-m} + C(h|\log h|)^{1-m}, \end{aligned}$$

which combined with (4.1) yields

$$\begin{aligned} \Vert \tilde{g} - g_h\Vert _{W^{1,1}(\Omega _h)} \le C'''K^{-1} \Vert \tilde{g} - g_h\Vert _{W^{1,1}(\Omega _h)} + CK^{m+N/2}h^{1-m} + C(h|\log h|)^{1-m}. \end{aligned}$$

If $K \ge 2C'''$, then this implies the desired estimate (3.2), which together with Proposition completes the proof of Theorem .

7 Numerical example

Letting $\Omega = \{(x, y) \in \mathbb {R}^2{:}\, \frac{(x-0.12)^2}{4} + \frac{(y+0.2)^2}{9} < 1, \; (x-0.7)^2 + (y-0.1)^2 > 0.5^2\}$, which is non-convex, we set an exact solution to be $u(x, y) = x^2$. We define f and $\tau$ so that (1.1) holds. They have natural extensions to $\mathbb {R}^2$, which are exploited as $\tilde{f}$ and $\tilde{\tau }$. Then we compute approximate solutions $u_h^k$ of (1.2) based on the $P_k$-finite elements ($k=1,2,3$), using the software FreeFEM++ [11]. The errors $\Vert u - u_h^k\Vert _{L^\infty (\Omega _h)}$ and $\Vert \nabla (u - u_h^k)\Vert _{L^\infty (\Omega _h)}$, which are calculated with the use of $P_4$-finite element spaces, are reported in Tables 1 and 2, respectively.

Table 1 Behavior of the $L^\infty$-errors for the $P_k$-approximation ($k=1,2,3$)

Table 2 Behavior of the $W^{1,\infty }$-errors for the $P_k$-approximation ($k=1,2,3$)

We see that the result for $k = 1$ is in accordance with Theorem . The one for $k = 3$ (although it is not covered by our theory) is also consistent with our theoretical expectation made in Remark (iii). When $k = 2$, the $L^\infty$-error remains sub-optimal convergence as expected. However, the $W^{1,\infty }$-error seems to be $O(h^2)$, which is significantly better than in the $P_3$-case. We remark that such behavior was also observed for different (and apparently more complicated) choices of $\Omega$ and u. There might be a super-convergence phenomenon in the $P_2$-approximation for Neumann problems in 2D smooth domains.

Remark 7.1

If $k \ge 2$ and $\tilde{\tau }$ is chosen as $\nabla u\cdot n_h$, then $u_h^k$ agrees with u (note that the above u is quadratic), because this amounts to assuming that the original problem (1.1) is given in a polygon $\Omega _h$. This was observed in our numerical experiment as well (up to rounding errors). However, since such $\tilde{\tau }$ is unavailable without knowing an exact solution, one cannot expect it in a practical computation.

Appendices

Appendix A: Auxiliary boundary-skin estimates

1.1 Local coordinate representation

We exploit the notations and observations given in [12, Section 8], which we briefly describe here. Since $\Omega$ is a bounded $C^\infty$-domain, there exist a system of local coordinates $\{(U_r, y_r, \varphi _r)\}_{r=1}^M$ such that $\{U_r\}_{r=1}^M$ forms an open covering of $\Gamma$, $y_r = (y_r', y_{rN})$ is a rotated coordinate of x, and $\varphi _r{:}\,\Delta _r \rightarrow \mathbb {R}$ gives a graph representation $\Phi _r(y_r') := (y_r', \varphi _r(y_r'))$ of $\Gamma \cap U_r$, where $\Delta _r$ is an open cube in $\mathbb {R}^{N-1}_{y_r'}$.

For $S\in \mathcal {S}_h$, we may assume that $S \cup \pi (S)$ is contained in some $U_r$, where $\pi {:}\,\Gamma (\delta _0) \rightarrow \Gamma$ is the projection to $\Gamma$ given in Sect. . Let $b_r{:}\,\mathbb {R}^N \rightarrow \mathbb {R}^{N-1}; y_r \mapsto y_r'$ be a projection to the base set and let $S' := b_r(\pi (S))$. Then $\Phi _r$ and $\Phi _{hr} := \pi ^*\circ \Phi _r$, where $\pi ^*{:}\,\Gamma \rightarrow \Gamma _h$ is the inverse map of $\pi |_{\Gamma _h}$, give smooth parameterizations of $\pi (S)$ and S respectively, with the domain $S'$. We also recall that $\pi ^*$ is also written as $\pi ^*(\Phi _r(y_r')) = \Phi _r(y_r') + t^*(\Phi _r(y_r')) n(\Phi _r(y_r'))$.

Let us represent integrals associated with S in terms of local coordinates. In what follows, we omit the subscript r for simplicity. First, surface integrals along $\pi (S)$ and S are expressed as

$$\begin{aligned} \int _{\pi (S)} f \, d\gamma =&\int _{S'} f(\Phi (y')) \sqrt{{\text {det}} G(y')} \, dy', \nonumber \\ \int _{S} f \, d\gamma _h =&\int _{S'} f(\Phi _h(y')) \sqrt{{\text {det}} G_h(y')} \, dy', \end{aligned}$$

where G and $G_h$ denote the Riemannian metric tensors obtained from the parameterizations $\Phi$ and $\Phi _h$, respectively. Next, let $\pi (S,\delta ) := \{\bar{x} + tn(\bar{x}){:}\, \bar{x} \in S, \; -\delta \le t\le \delta \}$ be a tubular neighborhood with the base $\pi (S)$, where $\delta = C_{0E}h^2$, and consider volume integrals over $\pi (S, \delta )$. For this we introduce a one-to-one transformation $\Psi {:}\,S'\times [-\delta , \delta ] \rightarrow \pi (S, \delta )$ by

$$\begin{aligned} y = \Psi (z', t) := \Phi (z') + t n(\Phi (z')) \Longleftrightarrow z' = b(\pi (y)), \; t = d(y). \end{aligned}$$

Then, by change of variables, we obtain

$$\begin{aligned} \int _{\pi (S,\delta )} f(y) \, dy = \int _{S'\times [-\delta , \delta ]} f(\Psi (z', t)) {\text {det}} J(z', t) \, dz' dt, \end{aligned}$$

where $J := \nabla _{(z', t)} \Psi$ denotes the Jacobi matrix of $\Psi$. In the formulas above, ${\text {det}}G$, ${\text {det}}G_h$, and ${\text {det}}J$ can be bounded, from above and below, by positive constants depending on the $C^{1,1}$-regularity of $\Omega$, provided h is sufficiently small (for the proof, see [12, Section 8]).

1.2 Proof of (2.2)

In [12, Theorem 8.3], we estimated the $L^p$-norm of a function in the full layer $\Gamma (\delta )$. By slightly modifying the proof there, we can estimate it in $\Omega _h{\setminus }\Omega$, which is important to dispense with extensions from $\Omega _h$ to $\tilde{\Omega }$.

Lemma A.1

Let $f \in W^{1,p}(\Omega _h) \, (1\le p\le \infty )$ and $\delta = C_{0E}h^2$. Then we have

$$\begin{aligned} \Vert f\Vert _{L^p(\Omega _h{\setminus }\Omega )} \le C( \delta ^{1/p} \Vert f\Vert _{L^p(\Gamma _h)} + \delta \Vert (n\circ \pi ) \cdot \nabla f\Vert _{L^p(\Omega _h{\setminus }\Omega )} ), \end{aligned}$$

where C is independent of $\delta$ and f.

Proof

To simplify the notation we use the abbreviation $t^*(z')$ to imply $t^*(\Phi (z'))$. For each $S \in \mathcal {S}_h$ we observe that

$$\begin{aligned}&\int _{(\Omega _h{\setminus }\Omega ) \cap \pi (S,\delta )} |f(y)|^p\,dy\\&\quad = \int _{S'} \int _0^{\max \{0, t^*(z')\}} |f(\Psi (z', t))|^p {\text {det}} J \, dt \, dz' \\&\quad \le C \int _{S'} \int _0^{\max \{0, t^*(z')\}} \Big ( |f(\Phi _h(z'))|^p + |f(\Psi (z', t)) - f(\Phi _h(z'))|^p \Big ) \, dt \, dz' \\&\quad =: I_1 + I_2, \end{aligned}$$

and that for $z' \in S'$ and $0\le t\le t^*(z')$

$$\begin{aligned} |f(\Psi (z', t)) - f(\Phi _h(z'))|&= \left| \int _t^{t^*(z')} n(\Phi (z'))\cdot \nabla f(\Psi (z', s)) \, ds \right| \\&\le \int _0^{t^*(z')} |n(\Phi (z')) \cdot \nabla f(\Psi (z', s))| \, ds \\&\le t^*(z')^{1- 1/p} \left( \int _0^{t^*(z')} |n(\Phi (z')) \cdot \nabla f(\Psi (z', s))|^p \, ds \right) ^{1/p}. \end{aligned}$$

Then it follows that

$$\begin{aligned}&I_1 \le C\Vert t^*\Vert _{L^\infty (S)} \int _{S'} |f(\Phi _h(z'))|^p\, dz'\\&\quad \le C\delta \int _{S'} |f(\Phi _h(z'))|^p \sqrt{{\text {det}}G_h}\,dz' = C\delta \Vert f\Vert _{L^p(S)}^p \end{aligned}$$

and that

$$\begin{aligned} I_2&\le C\Vert t^*\Vert _{L^\infty (S)}^p \int _{S'} \int _0^{\max \{0, t^*(z')\}} |n(\Phi (z')) \cdot \nabla f(\Psi (z', s))|^p {\text {det}} J \, dt \, dz' \\&\le C\delta ^p \Vert n\circ \pi \cdot \nabla f\Vert _{L^p(\pi (S,\delta ))}^p. \end{aligned}$$

Adding up the above estimates for $S \in \mathcal {S}_h$ gives the conclusion. $\square$

Lemma A.2

For a measurable set $D \subset \mathbb {R}^N$ and $f \in W^{1,\infty }(\Gamma (\delta ))$ we have

$$\begin{aligned} \Vert f - f\circ \pi \Vert _{L^\infty (\Gamma (\delta ) \cap D)} \le \delta \Vert \nabla f\Vert _{L^\infty (\Gamma (\delta ) \cap D_{2\delta })}, \end{aligned}$$

where $D_{2\delta } = \{x\in \mathbb {R}^N{:}\, {\text {dist}}(x, D) \le 2\delta \}$.

Proof

This is an easy consequence of the Lipschitz continuity of f. $\square$

1.3 Proof of Proposition

Let us prove stability properties of the extension operator P defined in Sect. .

Theorem A.1

Let $f \in W^{k,p}(\Omega )$ with $k=0,1,2$, and $p \in [1, \infty ]$. Then we have

$$\begin{aligned} \Vert Pf\Vert _{W^{k, p}(\Gamma (\delta ))}&\le C\Vert f\Vert _{W^{k, p}(\Omega \cap \Gamma (2\delta ))}, \end{aligned}$$

where C is independent of $\delta$ and f.

Proof

First, for each $S \in \mathcal {S}_h$ we show

$$\begin{aligned} \Vert Pf\Vert _{L^p(\pi (S,\delta ) {\setminus } \Omega )}^p \le C \Vert f\Vert _{L^p(\pi (S, 2\delta ) \cap \Omega )}^p. \end{aligned}$$

In fact we have

$$\begin{aligned} \int _{\pi (S,\delta ) {\setminus } \Omega } |Pf(y)|^p \, dy&\le C \int _{S' \times [0, \delta ]} |3f(z' - tn(z')) - 2f(z' - 2t n(z'))|^p \,dz'dt \\&\le C \int _{S' \times [0, \delta ]} \Big ( |f(z' - tn(z'))|^p + |f(z' - 2t n(z'))|^p \Big ) \,dz'dt \\&\le C \int _{\pi (S, \delta ) \cap \Omega } |f(y)|^p \,dy + C \int _{\pi (S, 2\delta ) \cap \Omega } |f(y)|^p \,dy. \end{aligned}$$

Next we show

$$\begin{aligned} \Vert \nabla Pf\Vert _{L^p(\pi (S,\delta ) {\setminus } \Omega )}^p \le C \Vert \nabla f\Vert _{L^p(\pi (S, 2\delta ) \cap \Omega )}^p. \end{aligned}$$

(A.1)

Since by the chain rule $\nabla _y = \nabla _y(b\circ \pi )\nabla _{z'} + (\nabla _yd)\partial _t$ and since $Pf(y) = 3f\circ \Psi (z', -t) - 2f\circ \Psi (z', -2t)$, it follows that

$$\begin{aligned} \nabla Pf(y)&= \nabla _y(b\circ \pi ) \Big ( 3\nabla _{z'}(f\circ \Psi )|_{(z',-t)} - 2\nabla _{z'}(f\circ \Psi )|_{(z',-2t)} \Big ) \nonumber \\&\qquad +\,\nabla _yd \Big ( -3\partial _t(f\circ \Psi )|_{(z',-t)} + 4\partial _t(f\circ \Psi )|_{(z',-2t)} \Big ), \quad y \in \pi (S,\delta ) {\setminus } \Omega . \end{aligned}$$

(A.2)

In particular, if $y \in \Gamma$ i.e. $t = 0$, then

$$\begin{aligned} \nabla Pf(y)= & {} \nabla _y(b\circ \pi ) \nabla _{z'}(f\circ \Psi )|_{(z',0)} + (\nabla _yd) \partial _t(f\circ \Psi )|_{(z', 0)}\\= & {} J^{-1}(z', 0) J(z', 0)\nabla _yf(y) = \nabla f(y), \end{aligned}$$

which ensures that $Pf(y) \in W^{2, p}(\pi (S, \delta ))$. Now, noting that $\nabla _y { \left( {\begin{matrix} b\circ \pi \\ d \end{matrix}}\right) } = J^{-1}(z', t)$ and that $\nabla _{(z', t)}(f\circ \Psi )|_{(z', -it)} = J(z', -it) (\nabla _y f)|_{\Psi (z', -it)} \, (i=1,2)$ where J and $J^{-1}$ depend on the $C^{1,1}$-regularity of $\Omega$, we deduce that

$$\begin{aligned} \int _{\pi (S,\delta ) {\setminus } \Omega } |\nabla Pf(y)|^p \, dy \le C \int _{S'\times [0, \delta ]} \Big ( \big | (\nabla _yf)|_{\Psi (z', -t)} \big |^p + \big | (\nabla _yf)|_{\Psi (z', -2t)} \big |^p \Big ) \, dz'dt, \end{aligned}$$

from which (A.1) follows.

Finally we show

$$\begin{aligned} \Vert \nabla ^2 Pf\Vert _{L^p(\pi (S,\delta ) {\setminus } \Omega )}^p \le C (\Vert \nabla ^2 f\Vert _{L^p(\pi (S, 2\delta ) \cap \Omega )}^p + \Vert \nabla f\Vert _{L^p(\pi (S, 2\delta ) \cap \Omega )}^p). \end{aligned}$$

(A.3)

By differentiating (A.2) we find that for $y \in \pi (S, \delta ) {\setminus } \Omega$

$$\begin{aligned} \nabla ^2Pf(y) = \sum _{i=1}^2 \Big ( A_i(z', t) \nabla _{(z', t)}^2(f\circ \Psi )_{(z', -it)} + B_i(z', t) \nabla _{(z', t)}(f\circ \Psi )_{(z', -it)} \Big ), \end{aligned}$$

where the coefficient tensors $A_i, B_i$ depend on the $C^{1,1}$-regularity of $\Omega$. Then the $L^p$-norm of the above quantity can be estimated similarly as before and one obtains (A.3).

Adding up the above estimates for $S \in \mathcal {S}_h$ deduces the desired stability properties. $\square$

We also need local stability of the extension operator as follows.

Corollary A.1

For a measurable set $D \subset \mathbb {R}^N$ and $\delta = C_{0E}h^2$ we have

$$\begin{aligned} \Vert Pf\Vert _{W^{k, \infty }(\Gamma (\delta ) \cap D)} \le C\Vert f\Vert _{W^{k, \infty }(\Omega \cap \Gamma (2\delta ) \cap D_{3\delta })} \quad (k=0,1,2), \end{aligned}$$

where $D_{3\delta } = \{x\in \mathbb {R}^N{:}\, {\text {dist}}(x, D) \le 3\delta \}$ and C is independent of $\delta$, f, and D.

Proof

We address the $L^\infty$-norm of $\nabla Pf$; the treatment of Pf and $\nabla ^2Pf$ is similar. For each $S \in \mathcal {S}_h$, we find from the analysis of Theorem that $\nabla Pf(y)$ for $y \in \pi (S, \delta ) {\setminus } \Omega$ can be expressed as

$$\begin{aligned} \nabla Pf(y) = \sum _{i=1}^2 A_i(z', t) (\nabla _yf)|_{\Psi (z', -it)}, \end{aligned}$$

where the matrices $A_i$ depend on the $C^{0,1}$-regularity of $\Omega$. Then the desired estimate follows from the observation that if $y = \Psi (z', t) \in \pi (S, \delta ) \cap D {\setminus } \Omega$ then $\Psi (z', -it) \in \pi (S, i\delta ) \cap D_{3\delta } \cap \Omega$ for $i=1,2$. $\square$

Appendix B: Analysis of regularized Green’s functions

1.1 Estimates for $\tilde{g}$

Recall that for arbitrarily fixed $x_0 \in \Omega _h$ we have introduced $\eta \in C_0^\infty (\Omega _h \cap \Omega )$ and $g_m \in C^\infty (\overline{\Omega }) \, (m=0,1)$ in Sect. . Using the Green’s function G(x, y) for the operator $-\,\Delta + 1$ in $\Omega$ with the homogeneous Neumann boundary condition, one can represent $g_m$ as

$$\begin{aligned} g_0(x) = \int _{{\text {supp}}\eta } G(x, y)\eta (y) \, dy, \quad g_1(x) = -\int _{{\text {supp}}\eta } \partial _yG(x, y)\eta (y) \, dy, \quad x \in \Omega . \end{aligned}$$

The following derivative estimates for G are well known (see e.g. [13, p. 965]):

$$\begin{aligned} |\nabla _x^k \nabla _y^l G(x, y)| \le {\left\{ \begin{array}{ll} C(1 + |x - y|^{2 - l - k - N}) &{}\quad (l+k+N > 2), \\ C(1 + \big |\log |x - y|\big |) &{}\quad (N=2, l=k=0). \end{array}\right. } \end{aligned}$$

From this, combined with a dyadic decomposition of $\Omega$, we derive some local and global estimates for $g_m$ and its extension $\tilde{g}_m := Pg_m$. Below the subscript m will be dropped for simplicity.

Lemma B.1

Let $\mathcal {A}_{\Omega _h}(x_0, d_0) = \{\Omega _h \cap A_j\}_{j=0}^J$ be a dyadic decomposition of $\Omega _h$ with $d_0 \in [4h, 1]$. Then, for $j=1, \dots , J$ and $k\ge 0$ we have

$$\begin{aligned} \Vert \nabla ^k g\Vert _{L^\infty (\Omega \cap A_j)} \le {\left\{ \begin{array}{ll} C(1 + d_j^{2-m-k-N}) &{}\quad (m+k+N>2), \\ C(1 + |\log d_j|) &{}\quad (N=2, m=k=0), \end{array}\right. } \end{aligned}$$

where C is independent of $x_0, d_0, h, j$, and $\partial$.

Proof

We only consider $m+k+N>2$ because the other case can be treated similarly. Notice that if $x \in \Omega \cap A_j \, (j\ge 1)$ and $y \in {\text {supp}}\eta$ then $|x - y| \ge \frac{3}{4} d_{j-1}$, which is obtained from $|x - x_0| \ge d_{j-1}$ and $|y - x_0|\le h$. It then follows that

$$\begin{aligned} \Vert \nabla ^k g\Vert _{L^\infty (\Omega \cap A_j)}&= \sup _{x\in \Omega \cap A_j} \left| \int _{{\text {supp}} \eta } \partial _y^m \nabla _x^kG(x, y) \eta (y) \, dy \right| \\&\le \sup _{|x - y|\ge \frac{3}{4} d_{j-1} |\partial _y^m\nabla _x^kG(x, y)|}\\&\le C(1 + d_j^{2-m-k-N}), \end{aligned}$$

which completes the proof. $\square$

We transfer these estimates in $\Omega$ to those in $\tilde{\Omega }= \Omega \cup \Gamma (\delta )$ using an extension operator and its stability.

Lemma B.2

Let $\mathcal {A}_{\Omega _h}(x_0, d_0) = \{\Omega _h \cap A_j\}_{j=0}^J$ be a dyadic decomposition of $\Omega _h$ with $d_0 \in [h, 1]$, $\delta = C_{0E}h^2$. For $p \in [1,\infty ]$, $j = 1, \dots , J$, and $m = 0,1$, we have

$$\begin{aligned} \Vert \nabla ^2 \tilde{g}\Vert _{L^p(\tilde{\Omega }\cap A_j)} \le Cd_j^{-m-N/p'}, \end{aligned}$$

where $p' = p/(p-1)$ and C is independent of $x_0, d_0, h, j$, and $\partial$.

Proof

By the Hölder inequality and Lemma we see that

$$\begin{aligned} \Vert \nabla ^2 \tilde{g}\Vert _{L^p(\tilde{\Omega }\cap A_j)}&\le C |\Omega _h \cap A_j|^{1/p} \Vert \nabla ^2 \tilde{g}\Vert _{L^\infty (\tilde{\Omega }\cap A_j)} \le Cd_j^{N/p} \Vert g\Vert _{W^{2, \infty }(\Omega \cap A_j^{(1/4)})} \\&\le Cd_j^{N/p} (1 + d_j^{2-m-N} + d_j^{1-m-N} + d_j^{-m-N}) \le Cd_j^{-m-N/p'}, \end{aligned}$$

where we have used $d_j \le 2 {\text {diam}}\,\Omega$ in the last inequality. $\square$

We also need local estimates in intersections of annuli and boundary-skins (or boundaries).

Lemma B.3

Under the assumptions in Lemma , let $k=0,1,2$. Then we have

$$\begin{aligned} \Vert \nabla ^k \tilde{g}\Vert _{L^p(\Gamma (\delta ) \cap A_j)}&\le C (\delta d_j^{N-1})^{1/p} (1 + d_j^{2 - m - k - N}), \\ \Vert \nabla ^k g\Vert _{L^p(\Gamma \cap A_j)} + \Vert \nabla ^k \tilde{g}\Vert _{L^p(\Gamma _h \cap A_j)}&\le C d_j^{(N-1)/p} (1 + d_j^{2 - m - k - N}), \end{aligned}$$

provided $m+k+N > 2$. Even when $N=2$ and $m=k=0$, the above estimates hold with the factor $d_j^{2-m-k-N}$ replaced by $|\log d_j|$. The constants C are independent of $x_0, d_0, h, j$, and $\partial$.

Proof

We only consider $m+k+N > 2$ since the other case may be treated similarly. From Corollary and Lemma we deduce that (note that $(A_j)_{3\delta } \subset A_j^{(1/4)}$ for small h)

$$\begin{aligned} \Vert \nabla ^k \tilde{g}\Vert _{L^p(\Gamma (\delta ) \cap A_j)}&\le |\Gamma (\delta ) \cap A_j|^{1/p} \Vert \nabla ^k \tilde{g}\Vert _{L^\infty (\Gamma (\delta ) \cap A_j)} \\&\le C(\delta d_j^{N-1})^{1/p} \Vert g\Vert _{W^{k,\infty }(\Omega \cap \Gamma (2\delta ) \cap A_j^{(1/4)})} \\&\le C(\delta d_j^{N-1})^{1/p} (1 + d_j^{2-m-k-N}), \end{aligned}$$

where we have used $d_j \le 2 {\text {diam}}\,\Omega$ in the second line. Similarly,

$$\begin{aligned} \Vert \nabla ^k \tilde{g}\Vert _{L^p(\Gamma _h \cap A_j)}&\le |\Gamma _h \cap A_j|^{1/p} \Vert \nabla ^k \tilde{g}\Vert _{L^\infty (\Gamma _h \cap A_j)} \\&\le Cd_j^{(N-1)/p} \Vert g\Vert _{W^{k,\infty }(\Omega \cap \Gamma (2\delta ) \cap A_j^{(1/4)})} \\&\le Cd_j^{(N-1)/p}(1 + d_j^{2-m-k-N}). \end{aligned}$$

One sees that $\Vert \nabla ^k g\Vert _{L^p(\Gamma \cap A_j)}$ obeys the same estimate. $\square$

Remark B.1

The three lemmas above remain true with $A_j$ replaced by $A_j^{(s)} (0\le s < 1)$, where the constants C become dependent on the choice of s.

Especially when $p=1$, the following global estimate in a boundary-skin layer holds.

Corollary B.1

Let $\delta = C_{0E}h^2$ with sufficiently small h. Then we have

$$\begin{aligned}&\Vert \tilde{g}_0\Vert _{W^{k, 1}(\Gamma (\delta ))} \le {\left\{ \begin{array}{ll} C\delta &{}\quad (k=0), \\ C\delta |\log h| &{}\quad (k=1), \\ C\delta h^{-1} &{}\quad (k=2), \end{array}\right. }\\&\Vert \nabla ^k g_0\Vert _{L^1(\Gamma )} + \Vert \nabla ^k \tilde{g}_0\Vert _{L^1(\Gamma _h)} \le {\left\{ \begin{array}{ll} C &{}\quad (k=0), \\ C|\log h| &{}\quad (k=1), \\ Ch^{-1} &{}\quad (k=2), \end{array}\right. } \end{aligned}$$

and

$$\begin{aligned}&\Vert \tilde{g}_1\Vert _{W^{k, 1}(\Gamma (\delta ))} \le {\left\{ \begin{array}{ll} C\delta |\log h| &{}\quad (k=0), \\ C\delta h^{-1} &{}\quad (k=1), \\ C\delta h^{-2} &{}\quad (k=2), \end{array}\right. }\\&\Vert \nabla ^k g_1\Vert _{L^1(\Gamma )} + \Vert \nabla ^k \tilde{g}_1\Vert _{L^1(\Gamma _h)} \le {\left\{ \begin{array}{ll} C|\log h| &{}\quad (k=0), \\ Ch^{-1} &{}\quad (k=1), \\ Ch^{-2} &{}\quad (k=2), \end{array}\right. } \end{aligned}$$

where C is independent of $x_0, h$, and $\partial$.

Proof

We only consider the estimates in $W^{k,1}(\Gamma (\delta ))$ because the boundary estimates can be derived similarly. With a dyadic decomposition $\mathcal {A}_{\Omega _h}(x_0, 4h) = \{\Omega _h \cap A_j\}_{j=0}^J$, we compute $\sum _{j=0}^J \Vert \tilde{g}\Vert _{W^{k,1}(\Gamma (\delta ) \cap A_j)}$. When $j \ge 1$, it follows from Lemma that

$$\begin{aligned} \Vert \tilde{g}\Vert _{W^{k, 1}(\Gamma (\delta ) \cap A_j)} \le {\left\{ \begin{array}{ll} C(\delta d_j^{N-1}) d_j^{2-m-k-N} &{}\quad (m+k+N>2), \\ C(\delta d_j^{N-1}) |\log d_j| &{}\quad (N=2, m=k=0). \end{array}\right. } \end{aligned}$$

(B.1)

When $j = 0$, notice that ${\text {dist}}({\text {supp}} \eta , \Gamma (2\delta )) \ge Ch = \frac{C}{4} d_0$ for sufficiently small h, which results from (3.1). Then, calculating in the same way as above, we find that (B.1) holds for $j=0$ as well. Adding up the above estimate for $j = 0, \dots , J$ and using (2.5), we obtain the desired result. $\square$

Remark B.2

We could improve the above estimates for $g_0$ when $k=1$ if the Dirichlet boundary condition were considered. In fact, the Green’s function $G_D(x, y)$ in this case is known to satisfy $|\nabla _x G_D(x, y)| \le C{\text {dist}}(y, \partial \Omega ) |x - y|^{-N}$ (see [10, Theorem 3.3(v)]). Then, taking a dyadic decomposition with $d_0 = {\text {dist}}( {\text {supp}}\eta , \partial \Omega ) \ge Ch$, we see that

$$\begin{aligned} \Vert \nabla \tilde{g}_0\Vert _{L^1(\Gamma _h)}\le & {} C\sum _{j=0}^J d_j^{N-1} \Vert \nabla \tilde{g}_0\Vert _{L^\infty (\Gamma _h \cap A_j)}\\\le & {} C {\text {dist}}( {\text {supp}}\eta , \partial \Omega ) \sum _{j=0}^J d_j^{-1} \nonumber \\\le & {} Cd_0 d_0^{-1} = C, \end{aligned}$$

and that $\Vert \nabla \tilde{g}_0\Vert _{L^1(\Gamma (\delta ))} \le C\delta$. However, such an auxiliary Green’s function estimate is not available in the case of the Neumann boundary condition. A similar inequality is proved in [17, eq. (5.8)] by a different method using the maximum principle, but its extension to the Neumann case seems non-trivial.

1.2 Estimates for $\tilde{w}$

Let us recall the situation of Sect. : fixing a dyadic decomposition $\mathcal {A}_{\Omega _h}(x_0, d_0)$ and an annulus $A_j \, (0\le j\le J)$, we have introduced the solution $w \in C^\infty (\overline{\Omega })$ of (5.2) for arbitrary $\varphi \in C_0^\infty (\Omega _h \cap A_j)$ such that $\Vert \varphi \Vert _{L^2(\Omega _h \cap A_j)} = 1$. Hence w is represented, using the Green’s function G(x, y), as

$$\begin{aligned} w(x) = \int _{\Omega \cap \Omega _h \cap A_j} G(x, y) \varphi (y) \, dy \quad (x \in \Omega ). \end{aligned}$$

Then we obtain the following local $L^\infty$-estimates away from $A_j$:

Lemma B.4

For $k=0,1,2$ and $\delta = C_{0E}h^2$, we have

$$\begin{aligned} \Vert \tilde{w}\Vert _{W^{k,\infty }(\tilde{\Omega }{\setminus } A_j^{(1/2)})} \le {\left\{ \begin{array}{ll} Cd_j^{2-k-N/2} &{}\quad (N+k>2), \\ Cd_j (1+|\log d_j|) &{}\quad (N=2, k=0), \end{array}\right. } \end{aligned}$$

where $\tilde{\Omega }:= \Omega \cup \Gamma (\delta )$, $\tilde{w} := Pw$, and C is independent of $h,x_0,d_0$, and j.

Proof

We focus on the case $N+k>2$; the other case is similar. We find that

$$\begin{aligned}&\Vert \tilde{w}\Vert _{W^{k,\infty }(\tilde{\Omega }{\setminus } A_j^{(1/2)})}\\&\quad \le C\Vert w\Vert _{W^{k,\infty }(\Omega {\setminus } A_j^{(1/4)})} = C\sum _{l=0}^k \sup _{x \in \Omega {\setminus } A_j^{(1/4)}} \left| \int _{\Omega \cap \Omega _h \cap A_j} \nabla _x^l G(x, y) \varphi (y) \, dy \right| \\&\quad \le C \sum _{l=0}^k |\Omega \cap \Omega _h \cap A_j|^{1/2} \sup _{|x - y| \ge d_{j-1}/8} |\nabla _x^l G(x, y)| \, \Vert \varphi \Vert _{L^2(\Omega \cap \Omega _h \cap A_j)} \\&\quad \le Cd_j^{N/2} \left( 1 + d_j^{2-N} + \cdots + d_j^{2-k-N}\right) \le Cd_j^{2-k-N/2}, \end{aligned}$$

where we have used $d_j \le 2 {\text {diam}}\,\Omega$ in the last inequality. $\square$

Remark B.3

The lemma remains true with $A_j^{(1/2)}$ replaced by $A_j^{(s)} \, (0<s\le 1)$, where the constant C becomes dependent on the choice of s.