Mortar Coupling of hp-Discontinuous Galerkin and Boundary Element Methods for the Helmholtz Equation

Abstract

We design and analyze a coupling of a discontinuous Galerkin finite element method with a boundary element method to solve the Helmholtz equation with variable coefficients in three dimensions. The coupling is realized with a mortar variable that is related to an impedance trace on a smooth interface. The method obtained has a block structure with nonsingular subblocks. We prove quasi-optimality of the $h$- and $p$-versions of the scheme, under a threshold condition on the approximability properties of the discrete spaces. Amongst others, an essential tool in the analysis is a novel discontinuous-to-continuous reconstruction operator on tetrahedral meshes with curved faces.

Appendices

Consistency of Method (3.12)

Proof of Lemma 1

Proving assertion (3.15) is equivalent to proving that the continuous solution $(u,m,u^{ext})$ solves also the three equations in (3.12). Since $u\in H^{\frac{3}{2}+t}(\varOmega )$ we have that

$$\begin{aligned} \llbracket u \rrbracket =0, \qquad \llbracket \nabla u\rrbracket =0, \qquad \{\!\!\!\{\nabla u\}\!\!\!\}=\nabla u \qquad \text {on }{\mathcal {F}}_h^I. \end{aligned}$$

(A.1)

We multiply (2.6) by ${\overline{v_h}}\in V_h$ and integrate elementwise by parts to get

$$\begin{aligned} \sum _{K\in \varOmega _h} \left( -\int _{\partial K} \nu \nabla u \cdot {\mathbf {n}}_{\varGamma }{\overline{v_h}} +\int _{K}\nu \,\nabla u \cdot \overline{\nabla v_h}\right) -\int _\varOmega (k n)^2 u {\overline{v_h}} =\int _\varOmega f{\overline{v_h}} . \end{aligned}$$

With the aid of the boundary condition in (2.6), inserting the parameter $\delta$, and using the fact that $\nu =1$ on $\varGamma$, we manipulate the boundary term as follows:

$$\begin{aligned}&-\sum _{K\in \varOmega _h} \int _{\partial K} \nu \nabla u \cdot {\mathbf {n}}_{\varGamma }{\overline{v_h}} \\&\quad = -\int _{\varGamma }\delta \nabla u\cdot {\mathbf {n}}_{\varGamma }{\overline{v_h}} -\int _{\varGamma }(1-\delta ) m{\overline{v_h}} +\int _{\varGamma } \text {i}k(1-\delta )u {\overline{v_h}} -\int _{{\mathcal {F}}_h^I}\nu \nabla u \cdot \llbracket {\overline{v_h}}\rrbracket \\&\qquad +\int _{\varGamma }(\text {i}k)^{-1}\delta m\overline{\nabla v_h\cdot {\mathbf {n}}_{\varGamma }} -\int _{\varGamma }(\text {i}k)^{-1}\delta \nabla u\cdot {\mathbf {n}}_{\varGamma }\overline{\nabla v_h\cdot {\mathbf {n}}_{\varGamma }} -\int _{\varGamma }\delta u \overline{\nabla v_h\cdot {\mathbf {n}}_{\varGamma }} . \end{aligned}$$

Properties (A.1) and the above identity lead to the consistency of the first equation of (3.12), i.e.,

$$\begin{aligned}&\sum _{K\in \varOmega _h} a^K_h(u,v_h) + b_h^\varGamma (u,v_h) - (m, \delta (\text {i}k)^{-1} \nabla _h v_h\cdot {\mathbf {n}}_{\varGamma }+ (1-\delta )v_h)_{0,\varGamma }\\&\quad = (f,v_h)_{0,\varOmega } \qquad \forall v_h\in V_h. \end{aligned}$$

To show the consistency of the second equation of (3.12), we multiply (2.10), which is an equivalent formulation of (2.7), by ${\overline{v_h^{\text {ext}}}}\in Z_h$ and integrate over $\varGamma$:

$$\begin{aligned} \langle ({\mathcal {B}}_k+ \text {i}k{\mathcal {A}}'_k) u^{ext}- {\mathcal {A}}'_km, v_h^{\text {ext}}\rangle =0 \qquad \forall v_h^{\text {ext}}\in Z_h. \end{aligned}$$

Eventually, multiplying (2.8) by ${\overline{\lambda _h}}\in W_h$ and integrating over $\varGamma$, we get

$$\begin{aligned} \langle u , \lambda _h\rangle - \left\langle \left( \frac{1}{2} + {\mathcal {K}}_k\right) u^{ext}- {\mathcal {V}}_k(m- iku^{ext}), \lambda _h\right\rangle = 0. \end{aligned}$$

Similarly as above, the boundary condition in (2.6) leads to

$$\begin{aligned} \langle - \delta (\text {i}k)^{-1}\nabla u\cdot {\mathbf {n}}_{\varGamma },\lambda _h\rangle +\langle -\delta u,\lambda _h\rangle + \langle \delta (\text {i}k)^{-1}m,\lambda _h\rangle =0. \end{aligned}$$

Summing up the last two equations shows the consistency of the third equation in (3.12).

$\square$

An $hp$-Stable, Discontinuous-to-Continuous Reconstruction Operator on Curvilinear Simplicial Meshes

Here, we prove Theorem 2.

Let the mesh $\varOmega _h$ satisfy the shape regularity Assumption (3.1) and $v \in H_{{\text {pw}}}^1(\varOmega _h)$. We construct the operator ${\mathcal {P}}:H_{{\text {pw}}}^1(\varOmega _h) \rightarrow H^1(\varOmega )$ as the composition ${\mathcal {P}}:= {\mathcal {P}}_2\circ {\mathcal {P}}_1$ of two operators ${\mathcal {P}}_2$, ${\mathcal {P}}_1$ that we define below. Preliminarily, for each $K\in \varOmega _h$, we construct a quasi-uniform, shape regular simplicial decomposition ${{\widetilde{\varOmega }}}_h^K$ of $K$, such that the size of each element ${{\widetilde{K}}}$ of ${{\widetilde{\varOmega }}}_h^K$ is comparable to ${{\widetilde{h}}}_K:= h_K/\ell ^2$. Denote the union of all ${{\widetilde{\varOmega }}}_h^K$ by ${{\widetilde{\varOmega }}}_h$. By using a standard refinement strategy on the original mesh, we can additionally ensure that ${{\widetilde{\varOmega }}}_h$ does not contain hanging nodes. We also introduce

$$\begin{aligned} {{\widetilde{V}}}_h:=\{v \in \mathcal{S}^{1,0}(\varOmega ,{{\widetilde{\varOmega }}}_h)\,|\, v{}_{|_K} \in \mathcal{S}^{1,1}(K,{{\widetilde{\varOmega }}}_h^K) \quad \forall K \in \varOmega _h\}, \end{aligned}$$

(B.1)

the space of the mapped, piecewise linear polynomials over ${{\widetilde{\varOmega }}}_h$, which are continuous in each $K\in \varOmega _h$ but possibly discontinuous at the interfaces of $\varOmega _h$.

We define ${\mathcal {P}}_1: H_{{\text {pw}}}^1(\varOmega _h) \rightarrow {{\widetilde{V}}}_h$ as follows. For each $K\in \varOmega _h$, ${\mathcal {P}}_1(v{}_{|K})\in \mathcal{S}^{1,1}(K, {{\widetilde{\varOmega }}}_h^K)$ is the quasi-interpolant of v defined in [4, Sec. 4]. As for ${\mathcal {P}}_2: {{\widetilde{V}}}_h\rightarrow \mathcal{S}^{1,1}(\varOmega , {{\widetilde{\varOmega }}}_h)\subset H^1(\varOmega )$, we choose the lowest-order, Oswald-type operator introduced by Karakashian and Pascal in [37]. This operator interpolates the arithmetical averages of the degrees of freedom at each vertex of the mesh ${{\widetilde{\varOmega }}}_h$. Thus, we are actually going to prove Theorem 2 with ${\mathcal {P}}:H_{{\text {pw}}}^1(\varOmega _h) \rightarrow \mathcal{S}^{1,1}(\varOmega , {{\widetilde{\varOmega }}}_h)\subset H^1(\varOmega )$. For simplicity, throughout this section we assume that $h/{\ell ^2} \lesssim 1$ and $\ell \in {\mathbb {N}}$. The other cases follow similarly but would incur some cumbersome notation/case distinctions.

Before proving (4.13)–(4.15), we recall two propositions, which summarize the properties of the operators ${\mathcal {P}}_1$ and ${\mathcal {P}}_2$.

Proposition 5

For any element $K \in \varOmega _h$, the quasi-interpolant ${\mathcal {P}}_1:H_{{\text {pw}}}^1(\varOmega _h)\rightarrow {{\widetilde{V}}}_h$ satisfies the following estimates:

$$\begin{aligned} \Vert \nabla {\mathcal {P}}_1v \Vert _{0,K}&\lesssim \Vert \nabla v \Vert _{0,K} , \end{aligned}$$

(B.2)

$$\begin{aligned} \Vert v -{\mathcal {P}}_1v \Vert _{0,K}&\lesssim \Vert {{\mathfrak {h}}}\ell ^{-2} \nabla v \Vert _{0,K}, \end{aligned}$$

(B.3)

$$\begin{aligned} \Vert \llbracket {\mathcal {P}}_1v \rrbracket \Vert _{0,\partial K{\setminus } \varGamma }&\lesssim \Vert \llbracket v \rrbracket \Vert _{0,\partial K{\setminus } \varGamma }+ \Vert {{\mathfrak {h}}}^{1/2}\ell ^{-1} \nabla _h v \Vert _{0,\omega _K}, \end{aligned}$$

(B.4)

where $\omega _K$ in (B.4) denotes the set of elements sharing a face with K.

Proof

Bounds (B.2) and (B.3) follow from [4, Thm. 4.1] locally on K as the domain to obtain a function on the subtriangulation ${{\widetilde{\varOmega }}}_h^K$. We can apply [4, Thm. 4.1] since ${{\widetilde{\varOmega }}}_h^K$ fulfills (3.1) and thus (3.2), which is the condition required there.

To show (B.4), we fix a facet F shared by the elements K and $K'$. We get

$$\begin{aligned} \Vert \llbracket {\mathcal {P}}_1v \rrbracket \Vert _{0,F}&\le \Vert \llbracket v \rrbracket \Vert _{0,F} + \Vert \llbracket v - {\mathcal {P}}_1v \rrbracket \Vert _{0,F} \\&\le \Vert \llbracket v \rrbracket \Vert _{0,F} + \Vert \big (v - {\mathcal {P}}_1v \big )_{|K} \Vert _{0,F} + \Vert \big (v - {\mathcal {P}}_1v \big )_{|K'} \Vert _{0,F}. \end{aligned}$$

For brevity, we only consider the third term on the right-hand side. Transforming to the reference element, applying a multiplicative trace estimate and transforming back gives

$$\begin{aligned} \Vert \big (v - {\mathcal {P}}_1v \big )_{|K'} \Vert _{0,F}&\lesssim \Vert {{\mathfrak {h}}}^{-1/2} (v - {\mathcal {P}}_1v) \Vert _{0,K'} + \Vert v - {\mathcal {P}}_1v \Vert _{0,K'}^{1/2} \Vert \nabla (v - {\mathcal {P}}_1v) \Vert _{0,K'}^{1/2}. \end{aligned}$$

Inserting (B.2) and (B.3) yields (B.4). $\square$

Proposition 6

The Oswald-type operator ${\mathcal {P}}_2: {{\widetilde{V}}}_h\rightarrow \mathcal{S}^{1,1}(\varOmega , {{\widetilde{\varOmega }}}_h)$ satisfies the following properties:

$$\begin{aligned} \Vert {\widetilde{v}}_h - {\mathcal {P}}_2{\widetilde{v}}_h \Vert _{0,\varOmega } \lesssim \Vert {{\mathfrak {h}}}^{1/2} \ell ^{-1}\llbracket {\widetilde{v}}_h \rrbracket \Vert _{0,{\mathcal {F}}_h^I},\quad \Vert \nabla _h( {\widetilde{v}}_h - {\mathcal {P}}_2{\widetilde{v}}_h )\Vert _{0,\varOmega } \lesssim \Vert {{\mathfrak {h}}}^{-1/2} \ell \llbracket {\widetilde{v}}_h \rrbracket \Vert _{0,{\mathcal {F}}_h^I}. \end{aligned}$$

(B.5)

Proof

We claim that

$$\begin{aligned}&\Vert {\widetilde{v}}_h - {\mathcal {P}}_2{\widetilde{v}}_h \Vert _{0,\varOmega }^2 \lesssim \sum _{K\in \varOmega _h} {\Vert {{\widetilde{h}}}_{K}^{1/2}\llbracket {\widetilde{v}}_h \rrbracket \Vert _{0,\partial K{\setminus } \varGamma }^{2}} ,\quad \\&\Vert \nabla _h({\widetilde{v}}_h - {\mathcal {P}}_2{\widetilde{v}}_h) \Vert _{0,\varOmega }^2 \lesssim \sum _{K\in \varOmega _h} {\Vert {{\widetilde{h}}}_{K}^{-1/2}\llbracket {\widetilde{v}}_h \rrbracket \Vert _{0,\partial K{\setminus }\varGamma }^{2}. } \end{aligned}$$

This follows as in the proof of [37, Thm. 2.2], which only makes use of the definition of the Lagrangian degrees of freedom of ${\mathcal {P}}_2{\widetilde{v}}_h$ as arithmetical averages of the degrees of freedom of ${\widetilde{v}}_h$ and of the scaling properties of the basis functions. We remark that [37, Thm. 2.2] states the estimate in the $H^1$ seminorm; the estimate in the $L^2$ norm follows along the same lines; see also [5, Lemma 5.3]. Then, the estimates in (B.5) follow from the definition of ${{\widetilde{h}}}_K= h_K/\ell ^2$ and the fact that function ${\widetilde{v}}_h$ is continuous within each element $K \in \varOmega _h$, i.e., no extra jumps are introduced along the edges of the refined triangulation ${{\widetilde{\varOmega }}}_h$. $\square$

As an immediate consequence of the shape regularity of $\varOmega _h$ and the locality of the operator ${\mathcal {P}}_1$, we get

$$\begin{aligned}&\Vert {{\mathfrak {h}}}^{-1} \ell ^2\big ({\mathcal {P}}_1v - {\mathcal {P}}_2({\mathcal {P}}_1v) \big ) \Vert _{0,\varOmega }\nonumber \\&\quad \overset{(B.5)}{\lesssim } \Vert {{\mathfrak {h}}}^{-{1/2}} \ell \llbracket {\mathcal {P}}_1v \rrbracket \Vert _{0, {\mathcal {F}}_h^I}\overset{(B.4)}{\lesssim } \Vert {{\mathfrak {h}}}^{-{1/2}} \ell \llbracket v \rrbracket \Vert _{0, {\mathcal {F}}_h^I} + \Vert \nabla _hv \Vert _{0,\varOmega }. \end{aligned}$$

(B.6)

We prove further properties of the operator ${\mathcal {P}}_2$. First, proceeding as in Remark 2, we have the following inverse estimate for mapped, affine functions:

$$\begin{aligned} \Vert \nabla q \Vert _{0,{{\widetilde{K}}}} \lesssim {{\widetilde{h}}}_{K}^{-1} \Vert q \Vert _{0,{{\widetilde{K}}}} = \Vert {{\mathfrak {h}}}^{-1}\ell ^2 q \Vert _{0,{{\widetilde{K}}}} \quad \forall {{\widetilde{K}}}\in {{\widetilde{\varOmega }}}_h^K,\; \forall q \in \mathcal S^{1,1}\left( K, {{\widetilde{\varOmega }}}_h^K\right) . \end{aligned}$$

(B.7)

Next, we observe that

$$\begin{aligned} \Vert&\nabla _h{\mathcal {P}}_2({\mathcal {P}}_1v) \Vert _{0,\varOmega } \le \Vert \nabla _h({\mathcal {P}}_1v) \Vert _{0,\varOmega } + \Vert \nabla _h({\mathcal {P}}_1v - {\mathcal {P}}_2({\mathcal {P}}_1v)) \Vert _{0,\varOmega } \\&\overset{(B.2)}{\lesssim } \Vert \nabla _hv \Vert _{0,\varOmega } + \Vert \nabla _h({\mathcal {P}}_1v - {\mathcal {P}}_2({\mathcal {P}}_1v)) \Vert _{0,\varOmega }\\&\overset{(B.7)}{\lesssim } \Vert \nabla _hv \Vert _{0,\varOmega } + \Vert {{\mathfrak {h}}}^{-1}\ell ^2 \big ( {\mathcal {P}}_1v - {\mathcal {P}}_2({\mathcal {P}}_1v) \big ) \Vert _{0,\varOmega } \\&\overset{(B.6)}{\lesssim } \Vert \nabla _hv \Vert _{0,\varOmega } + \Vert {{\mathfrak {h}}}^{-1/2}\ell \llbracket v \rrbracket \Vert _{0,{\mathcal {F}}_h^I}. \end{aligned}$$

(B.8)

From this and the triangle inequality, we get (4.13).

In order to prove (4.14), we observe that the following approximation property of the operator ${\mathcal {P}}_2$ is valid:

$$\begin{aligned}&\Vert v - {\mathcal {P}}_2({\mathcal {P}}_1v) \Vert _{0,\varOmega } \le \Vert v - {\mathcal {P}}_1v \Vert _{0,\varOmega } + \Vert {\mathcal {P}}_1v - {\mathcal {P}}_2({\mathcal {P}}_1v) \Vert _{0,\varOmega }\\&\,\,\qquad \qquad \quad \qquad \overset{(B.3), (B.6)}{\lesssim } \Vert {{\mathfrak {h}}}\ell ^{-2}\nabla _hv \Vert _{0,\varOmega } + \Vert {{\mathfrak {h}}}^{1/2} \ell ^{-1} \llbracket v \rrbracket \Vert _{0, {\mathcal {F}}_h^I}. \end{aligned}$$

(B.9)

Then, (4.14) follows by the triangle inequality.

We are left to prove (4.15). To that end, we use a scaling argument. Given $v \in H_{{\text {pw}}}^1(\varOmega _h)$, for any $K\in \varOmega _h$, let ${{\widehat{v}}}$ be the polynomial pull-back of ${v}_{|_{K}}$ through the mapping $\varPhi _K: {\widehat{K}}\rightarrow K$. We denote the counterparts of ${\mathcal {P}}_1$ and ${\mathcal {P}}_2$ acting on the polynomials on ${\widehat{K}}$ by $\widehat{{\mathcal {P}}}_1$ and $\widehat{{\mathcal {P}}}_2$, respectively. For any boundary face $F\in {\mathcal {F}}_h^B$, we denote the pull-back of $F$ through $\varPhi _K$ by ${\widehat{F}}$, where $K$ is the unique element such that $F\subset \partial K$. For all $F\in {\mathcal {F}}_h^B$, we apply a scaling argument, the multiplicative trace inequality, and the Young inequality to get

$$\begin{aligned} \Vert v - {\mathcal {P}}_2({\mathcal {P}}_1v) \Vert _{0,F}^2&\lesssim \Vert {{\mathfrak {h}}}({{\widehat{v}}} - \widehat{{\mathcal {P}}}_2(\widehat{{\mathcal {P}}}_1{{\widehat{v}}}) )\Vert _{0,{\widehat{F}}}^2 \\&\lesssim \Vert {{\mathfrak {h}}}({{\widehat{v}}} - \widehat{{\mathcal {P}}}_2(\widehat{{\mathcal {P}}}_1\widehat{v})) \Vert _{0,{\widehat{K}}}^2\\&\quad + \Vert {{\mathfrak {h}}}({{\widehat{v}}} - \widehat{{\mathcal {P}}}_2(\widehat{{\mathcal {P}}}_1{{\widehat{v}}})) \Vert _{0,{\widehat{K}}} \Vert {{\mathfrak {h}}}{\widehat{\nabla }} ({{\widehat{v}}} - \widehat{{\mathcal {P}}}_2(\widehat{{\mathcal {P}}}_1{{\widehat{v}}})) \Vert _{0,{\widehat{K}}} \\&{\mathop {\lesssim }\limits ^{\ell \ge 1}} \Vert {{\mathfrak {h}}}\ell ({{\widehat{v}}} - \widehat{{\mathcal {P}}}_2(\widehat{{\mathcal {P}}}_1{{\widehat{v}}})) \Vert _{0,{\widehat{K}}}^2 + \Vert {{\mathfrak {h}}}\ell ^{-1} {\widehat{\nabla }} ({{\widehat{v}}} - \widehat{{\mathcal {P}}}_2(\widehat{{\mathcal {P}}}_1{{\widehat{v}}})) \Vert _{0,{\widehat{K}}}^2. \end{aligned}$$

Scaling back to $K$, summing over all the elements, and using the locality of the operators ${\mathcal {P}}_1$ and ${\mathcal {P}}_2$, as well the shape regularity of the meshes to insert the factor ${{\mathfrak {h}}}^{-{1/2}}\ell$, we deduce

$$\begin{aligned}&\Vert {{\mathfrak {h}}}^{-{1/2}} \ell \big (v - {\mathcal {P}}_2({\mathcal {P}}_1v) \big ) \Vert _{0,\varGamma }^2 \\&\quad \,\,\quad \lesssim \sum _{K\in \varOmega _h\text { with }{{\overline{K}}} \cap \partial \varOmega \in {\mathcal {F}}_h^B} \Vert {{\mathfrak {h}}}^{1/2} \ell ^2 ({{\widehat{v}}} - \widehat{{\mathcal {P}}}_2(\widehat{{\mathcal {P}}}_1{{\widehat{v}}})) \Vert _{0,{\widehat{K}}}^2 + \Vert {{\mathfrak {h}}}^{1/2}{\widehat{\nabla }} ({{\widehat{v}}} - \widehat{{\mathcal {P}}}_2(\widehat{{\mathcal {P}}}_1\widehat{v})) \Vert _{0,{\widehat{K}}}^2 \\&\quad \,\,\quad \lesssim \sum _{K\in \varOmega _h\text { with } {{\overline{K}}} \cap \partial \varOmega \in {\mathcal {F}}_h^B} \Vert {{\mathfrak {h}}}^{-1} \ell ^2 (v - {\mathcal {P}}_2({\mathcal {P}}_1v)) \Vert _{0,K}^2 + \Vert \nabla ( v - {\mathcal {P}}_2({\mathcal {P}}_1v)) \Vert _{0,K}^2\\&\, \overset{(B.9), (B.8)}{\lesssim } \Vert \nabla _hv \Vert _{0,\varOmega }^2 + \Vert {{\mathfrak {h}}}^{-1/2} \ell \llbracket v \rrbracket \Vert _{0, {\mathcal {F}}_h^I}^2, \end{aligned}$$

whence the assertion follows.

Explicit Error Estimates

Proof of Corollary 1

We start by noting that, for the special case $s = 1$, the arguments below show that Assumption 5 is valid with $\varepsilon = O(h/p)$. By Theorem 6, this fixes $\eta _0$.

To simplify the exposition, we restrict our attention to the case $p \ge s$. The case $p < s$ is a pure h-version that is shown along similar lines. We shall nevertheless write $\min (p,s) = s$ at the appropriate places.

By [4, Lemma 2.3], for any $v \in H^{s+1}(\varOmega )$, Assumption 7 implies that the following estimate for the pull-back ${\widehat{v}}:= v|_K \circ \varPhi _K$ is valid for all $K \in \varOmega _h$:

$$\begin{aligned} \Vert {\widehat{v}} \Vert _{s+1,{\widehat{K}}} \le c h_K^{s+1-3/2} \Vert v\Vert _{s+1,K}. \end{aligned}$$

(C.1)

We also note that, for $j \in \{0,1\}$ and for each face F of element K with corresponding pull-back ${\widehat{F}}:= \varPhi _K^{-1}(F)$, bounds (3.1) imply

$$\begin{aligned} |{{\widehat{v}}}|_{j,{\widehat{K}}} \sim h_K^{j-3/2} |v|_{j,K}, \qquad |{{\widehat{v}}}|_{0,{\widehat{F}}} \sim h_K^{-1} |v|_{0,F}, \qquad |{\widehat{\nabla }} {{\widehat{v}}}|_{0,{\widehat{F}}} \sim |\nabla v|_{0,F}. \end{aligned}$$

(C.2)

Properties (C.2) allow for transferring approximation results on the reference element ${\widehat{K}}$ to the physical elements K (“scaling argument”). The last preliminary ingredient are p-explicit approximation results on the reference element for which we refer, e.g., to [46, Lemma B.3, Thm. B.4]. As in, e.g., [44], combining the polynomial approximation results on ${\widehat{K}}$ with (C.2) and (C.1) allows for showing that

$$\begin{aligned} \inf _{v_h \in S^{p,0}(\varOmega ,\varOmega _h)} \Vert u - v_h\Vert _{\text {DG}^+(\varOmega )} \le c h^{\min (p,s)} p^{-s+\frac{1}{2}} \Vert u\Vert _{s+1,\varOmega }. \end{aligned}$$

(C.3)

For the approximation of $u^{ext}$ and $m$, we obviate the discussion of changes of variables in fractional Sobolev norms by resorting to appropriate liftings. For the approximation of $u^{ext}$, let $U^{ext} \in H^{s+1}(\varOmega )$ be a lifting of $u^{ext}$ with $\Vert U^{ext}\Vert _{s+1,\varOmega } \lesssim \Vert u^{ext}\Vert _{s+\frac{1}{2},\varGamma }$. Since the mesh $\varOmega _h$ is a regular mesh (see the discussion at the outset of Sect. 3.1), [46, Thm. B.4] provides an $H^1(\varOmega )$-conforming approximation with optimal convergence properties:

$$\begin{aligned} \inf _{v_h \in S^{p,1}(\varOmega ,\varOmega _h)} \Vert U^{ext} - v_h\Vert _{1,\varOmega } \le c h^{\min (p,s)} p^{-s} \Vert U^{ext}\Vert _{s+1,\varOmega } \le c h^{\min (p,s)}p^{-s} \Vert u^{ext}\Vert _{s+\frac{1}{2},\varGamma }. \end{aligned}$$

By taking the trace of $v_h$ on $\varGamma$, we obtain the desired approximation of $u^{ext}$. Finally, for $m$, let $M \in H^{s}(\varOmega )$ be a lifting of $m\in H^{s-\frac{1}{2}}(\varGamma )$ with $\Vert M\Vert _{H^s(\varOmega )} \lesssim \Vert m\Vert _{H^{s-\frac{1}{2}}(\varGamma )}$. Let $m_h \in S^{p-1,0}(\varGamma ,\varGamma _h)$ be the $L^2(\varGamma )$-projection of $m$ into $S^{p-1,0}(\varGamma ,\varGamma _h)$. For each face $F \in {\mathcal {F}}_h^B$, denote by $K_{F} \in \varOmega _h$ the element that has F as a face. Using approximation results on the reference element ${\widehat{K}}$ and the “scaling arguments” (C.2) we get

$$\begin{aligned} \Vert m- m_h\Vert _{0,F} \le c h_K^{\min (p,s)-1/2} p^{-s+1/2}\Vert M\Vert _{s,K_{F}}. \end{aligned}$$

(C.4)

By summation over all faces $F \in {\mathcal {F}}_h^B$, we arrive at

$$\begin{aligned} \Vert {{\mathfrak {h}}}^{1/2} p^{-1} (m-m_h)\Vert _{0,\varGamma } \lesssim h^{\min (p,s)} p^{-s-1/2}\Vert (m-m_h) \Vert _{s-\frac{1}{2},\varGamma }. \end{aligned}$$

The $H^{-\frac{1}{2}}(\varGamma )$-estimate is obtained by a standard duality argument using the orthogonality provided by the $L^2(\varGamma )$-projection:

$$\begin{aligned} \Vert m- m_h\Vert _{-\frac{1}{2}, \varGamma } = \sup _{v \in H^{\frac{1}{2}}(\varGamma )} \frac{|\langle m- m_h,v\rangle |}{\Vert v\Vert _{\frac{1}{2},\varGamma } } = \sup _{v \in H^{\frac{1}{2}}(\varGamma )} \inf _{v_h \in S^{p-1,0}(\varGamma ,\varGamma _h)} \frac{|\langle m- m_h,v - v_h\rangle | }{\Vert v\Vert _{\frac{1}{2},\varGamma } }. \end{aligned}$$

(C.5)

The infimum is estimated by taking $v_h$ as the $L^2(\varGamma )$-projection of v into $S^{p-1,0}(\varGamma ,\varGamma _h)$. To estimate $v - v_h$, let $V \in H^1(\varOmega )$ be a lifting of $v \in H^{\frac{1}{2}}(\varGamma )$ with $\Vert V\Vert _{1,\varOmega } \lesssim \Vert v\Vert _{\frac{1}{2},\varGamma }$. By the same arguments as in (C.4) (taking $s = 1$), we have

$$\begin{aligned} \Vert v - v_h\Vert _{0,F} \le c h_K^{\min (p,1)-1/2} p^{-1+1/2} \Vert V\Vert _{1,K_{F}}. \end{aligned}$$

Inserting this in (C.5) yields

$$\begin{aligned} \Vert m- m_h\Vert _{-\frac{1}{2},\varGamma }&\lesssim \sup _{v \in H^{\frac{1}{2}}(\varGamma )} \frac{ \sum _{F \in {\mathcal {F}}_h^B} \Vert m- m_h\Vert _{0,F} \Vert v - v_h\Vert _{0,F} }{\Vert v\Vert _{\frac{1}{2},\varGamma } } \\&\lesssim \sup _{v \in H^{\frac{1}{2}}(\varGamma )} \frac{1}{\Vert v\Vert _{\frac{1}{2},\varGamma }} \sum _{F \in {\mathcal {F}}_h^B} p^{-s} h_K^{\min (p,s)-1/2+1-1/2} \Vert M\Vert _{s,K_{F}}\Vert V\Vert _{1,K_{F}} \\&\lesssim h^{\min (p,s)} p^{-s} \Vert m\Vert _{s-\frac{1}{2},\varGamma } , \end{aligned}$$

which completes the proof. $\square$