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.

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.