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On the derivation of guaranteed and p-robust a posteriori error estimates for the Helmholtz equation

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Abstract

We propose a novel a posteriori error estimator for conforming finite element discretizations of two- and three-dimensional Helmholtz problems. The estimator is based on an equilibrated flux that is computed by solving patchwise mixed finite element problems. We show that the estimator is reliable up to a prefactor that tends to one with mesh refinement or with polynomial degree increase. We also derive a fully computable upper bound on the prefactor for several common settings of domains and boundary conditions. This leads to a guaranteed estimate without any assumption on the mesh size or the polynomial degree, though the obtained guaranteed bound may lead to large error overestimation. We next demonstrate that the estimator is locally efficient, robust in all regimes with respect to the polynomial degree, and asymptotically robust with respect to the wavenumber. Finally we present numerical experiments that illustrate our analysis and indicate that our theoretical results are sharp.

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Notes

  1. Our definition is slightly different from [44] as we include the wavenumber k in \({\sigma _{\mathrm{ba}}}\). This way, the quantity \({\sigma _{\mathrm{ba}}}\) is adimensional, and invariant under rescaling.

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Estimate on the best-approximation constant for boundary data

Estimate on the best-approximation constant for boundary data

In this appendix, we analyze the behavior of the quantity \({{\widetilde{\sigma }}}_{\mathrm{ba}}\) defined in (2.11) in terms of \({\sigma _{\mathrm{ba}}}\) defined in (2.7). For the sake of simplicity, in this section, the notation \(C({\widehat{\varOmega }},{\widehat{\varGamma }}_{\mathrm{D}})\) denotes a generic constant that only depends on the geometry of \(\varOmega \) and \({\varGamma _{\mathrm{D}}}\) but may vary from one occurrence to the other. In addition \(C_{\mathrm{qi}}(\kappa )\) is a “quasi-interpolation” constant that only depends on the mesh shape-regularity parameter \(\kappa \) [34, 43].

The results derived in this appendix rely on the following regularity assumption.

Assumption A.1

(Additional regularity) Let \(\phi \in L^2(\varOmega )\). We assume that if \(u \in H^1(\varOmega )\) satisfies

$$\begin{aligned} \left\{ \begin{array}{r@{\quad }l} -\varDelta u = \phi &{} \text { in } \varOmega , \\ u = 0 &{} \text { on } {\varGamma _{\mathrm{D}}}, \\ {\varvec{\nabla }}u {\cdot } {\varvec{n}}= 0 &{} \text { on } {\varGamma _{\mathrm{A}}}, \end{array} \right. \end{aligned}$$

then \(u \in H^2({\widetilde{\varOmega }})\) with

$$\begin{aligned} |u|_{2,{\widetilde{\varOmega }}} \le C({\widehat{\varOmega }},{\widehat{\varGamma }}_{\mathrm{D}}) \Vert \phi \Vert _{0,\varOmega }, \end{aligned}$$

where \({\widetilde{\varOmega }} \subset \varOmega \) is a neighborhood of \({\varGamma _{\mathrm{A}}}\) (i.e., \({\widetilde{\varOmega }}\) is an open subset of \(\varOmega \) and \({\varGamma _{\mathrm{A}}}\) is a subset of the closure of \({\widetilde{\varOmega }}\)) and the constant \(C({\widehat{\varOmega }},{\widehat{\varGamma }}_{\mathrm{D}})\) depends on the shape of \(\varOmega \) and the splitting of its boundary into \({\varGamma _{\mathrm{D}}}\) and \({\varGamma _{\mathrm{A}}}\) but not on its diameter \(h_\varOmega \). Furthermore, we assume that if \(\psi \in L^2({\varGamma _{\mathrm{A}}})\) and \(u \in H^1(\varOmega )\) solves

$$\begin{aligned} \left\{ \begin{array}{r@{\quad }l} -\varDelta u = 0 &{} \text { in } \varOmega , \\ u = 0 &{} \text { on } {\varGamma _{\mathrm{D}}}, \\ {\varvec{\nabla }}u {\cdot } {\varvec{n}}= \psi &{} \text { on } {\varGamma _{\mathrm{A}}}, \end{array} \right. \end{aligned}$$

then \(u \in H^{\frac{3}{2}}({\widetilde{\varOmega }})\) with

$$\begin{aligned} |u|_{\frac{3}{2},{\widetilde{\varOmega }}} \le C({\widehat{\varOmega }},{\widehat{\varGamma }}_{\mathrm{D}}) \Vert \psi \Vert _{0,{\varGamma _{\mathrm{A}}}}. \end{aligned}$$

Assumption A.1 is not an important restriction. Indeed, it is typically satisfied in applications, as the boundary \({\varGamma _{\mathrm{A}}}\) is artificially designed to enclose the region of interest. For instance, \({\varGamma _{\mathrm{A}}}\) is usually selected as the boundary of a convex polytope for scattering problems, so that Assumption A.1 holds (see [15] and [21, Lemma 1]). In the case of cavity problems, \({\varGamma _{\mathrm{A}}}\) is typically planar, and Assumption A.1 holds if the solid angle between \({\varGamma _{\mathrm{A}}}\) and \({\varGamma _{\mathrm{D}}}\) is less than or equal to \(\pi /2\) (we can perform an odd reflection across the Dirichlet boundary, and recover a situation similar to the scattering problem, see [15]). As a result, Assumption A.1 is satisfied in all the configurations depicted in Fig. 1.

Our next step is to employ a lifting operator \({\mathscr {L}}\) introduced in [44] that transforms the boundary right-hand side on \({\varGamma _{\mathrm{A}}}\), say \(\psi \), appearing in the definition (2.11) of \({{\widetilde{\sigma }}}_{\mathrm{ba}}\), into a volume right-hand side \({\mathscr {L}}_\psi \). We remark that there exists a function \(\chi \in C^\infty ({\overline{\varOmega }})\) such that \(0 \le \chi \le 1\) in \({\overline{\varOmega }}\), \(\chi = 0\) outside \({\widetilde{\varOmega }}\), and \(\chi = 1\) in a neighborhood on \({\varGamma _{\mathrm{A}}}\). In addition, a simple scaling argument shows that we can choose \(\chi \) such that

$$\begin{aligned} |\chi |_{j,\infty ,\varOmega } \le C({\widehat{\varOmega }},{\widehat{\varGamma }}_{\mathrm{D}}) h_\varOmega ^{-j} \end{aligned}$$

for all \(j \in {\mathbb {N}}\). The main novelty of the following result resides when both subsets \({\varGamma _{\mathrm{D}}}\) and \({\varGamma _{\mathrm{A}}}\) have positive measure and touch each other, so that only a regularity shift to \(H^{\frac{3}{2}}\) is available owing to Assumption A.1.

Lemma A.2

(Boundary lifting operator) Let Assumption A.1 hold. For all \(\psi \in L^2({\varGamma _{\mathrm{A}}})\), we define \({\mathscr {L}}_\psi \) as the unique element of \(H^1_{\varGamma _{\mathrm{D}}}(\varOmega )\) such that

$$\begin{aligned} a(w,{\mathscr {L}}_\psi ) = (w,\psi )_{\varGamma _{\mathrm{A}}}\end{aligned}$$
(A.1)

for all \(w \in H^1_{\varGamma _{\mathrm{D}}}(\varOmega )\), where

$$\begin{aligned} a(w,{\mathscr {L}}_\psi ) :=k^2(w,{\mathscr {L}}_\psi ) -ik(w,{\mathscr {L}}_\psi )_{\varGamma _{\mathrm{A}}}+ ({\varvec{\nabla }}w,{\varvec{\nabla }}{\mathscr {L}}_\psi ). \end{aligned}$$

Then we have

$$\begin{aligned} k\Vert {\mathscr {L}}_\psi \Vert _{0,{\varGamma _{\mathrm{A}}}}&\le \Vert \psi \Vert _{0,{\varGamma _{\mathrm{A}}}}, \end{aligned}$$
(A.2)
$$\begin{aligned} k^2 \Vert {\mathscr {L}}_\psi \Vert _{0,\varOmega }^2 + |{\mathscr {L}}_\psi |_{1,\varOmega }^2&\le \frac{1}{k} \Vert \psi \Vert _{0,{\varGamma _{\mathrm{A}}}}^2. \end{aligned}$$
(A.3)

In addition, we have

$$\begin{aligned} k^{\frac{1}{2}} \inf _{v_h \in V_h} |\chi {\mathscr {L}}_\psi - v_h|_{1,\varOmega } \le C({\widehat{\varOmega }},{\widehat{\varGamma }}_{\mathrm{D}}) C_{\mathrm{qi}}(\kappa ) \left( \left( \frac{kh}{p}\right) ^{\frac{1}{2}} + \frac{kh}{p} \right) \Vert \psi \Vert _{0,{\varGamma _{\mathrm{A}}}}, \end{aligned}$$
(A.4)

where the additional term \(\frac{kh}{p}\) is present only if \({\varGamma _{\mathrm{D}}}\) has positive surface measure.

Proof

We first pick \(w = {\mathscr {L}}_\psi \) as a test function in (A.1). Taking the imaginary part yields

$$\begin{aligned} k\Vert {\mathscr {L}}_\psi \Vert _{0,{\varGamma _{\mathrm{A}}}}^2 = -{\text {Im}}({\mathscr {L}}_\psi ,\psi )_{\varGamma _{\mathrm{A}}}\le \Vert \psi \Vert _{0,{\varGamma _{\mathrm{A}}}}\Vert {\mathscr {L}}_\psi \Vert _{0,{\varGamma _{\mathrm{A}}}}, \end{aligned}$$

and (A.2) follows. Now, we take the real part and use the above bound to obtain

$$\begin{aligned} k^2 \Vert {\mathscr {L}}_\psi \Vert _{0,\varOmega }^2 + |{\mathscr {L}}_\psi |_{1,\varOmega }^2 = {\text {Re}}({\mathscr {L}}_\psi ,\psi )_{{\varGamma _{\mathrm{A}}}} \le \frac{1}{k} \Vert \psi \Vert _{0,{\varGamma _{\mathrm{A}}}}^2. \end{aligned}$$

This yields (A.3). Finally, we observe that we can see \({\mathscr {L}}_\psi \) as the unique element of \(H^1_{\varGamma _{\mathrm{D}}}(\varOmega )\) such that

$$\begin{aligned} ({\varvec{\nabla }}w,{\varvec{\nabla }}{\mathscr {L}}_\psi ) = -k^2 (w,{\mathscr {L}}_\psi ) + (w,\psi -ik{\mathscr {L}}_\psi )_{{\varGamma _{\mathrm{A}}}} \end{aligned}$$

for all \(w \in H^1_{\varGamma _{\mathrm{D}}}(\varOmega )\).

At this point, we distinguish the case where \({\varGamma _{\mathrm{D}}}\) is of zero measure or not. If \(|{\varGamma _{\mathrm{D}}}| = 0\), then Assumption A.1 together with classical arguments (see [32]) away from \({\varGamma _{\mathrm{A}}}\) show that

$$\begin{aligned} \Vert {\mathscr {L}}_\psi \Vert _{\frac{3}{2},\varOmega }\le & {} C({\widehat{\varOmega }},{\widehat{\varGamma }}_{\mathrm{D}}) \left( k^2 \Vert {\mathscr {L}}_\psi \Vert _{\frac{1}{2},\varOmega } + \Vert \psi \Vert _{0,{\varGamma _{\mathrm{A}}}} + k\Vert {\mathscr {L}}_\psi \Vert _{0,{\varGamma _{\mathrm{A}}}} \right) \\\le & {} C({\widehat{\varOmega }},{\widehat{\varGamma }}_{\mathrm{D}}) \left( k^2 \sqrt{\Vert {\mathscr {L}}_\psi \Vert _{0,\varOmega }\Vert {\mathscr {L}}_\psi \Vert _{1,\varOmega }} + \Vert \psi \Vert _{0,{\varGamma _{\mathrm{A}}}} + k\Vert {\mathscr {L}}_\psi \Vert _{0,{\varGamma _{\mathrm{A}}}} \right) , \end{aligned}$$

and we conclude with (A.2) and (A.3) that

$$\begin{aligned} \Vert \chi {\mathscr {L}}_\psi \Vert _{\frac{3}{2},\varOmega } \le C({\widehat{\varOmega }},{\widehat{\varGamma }}_{\mathrm{D}}) \Vert {\mathscr {L}}_\psi \Vert _{\frac{3}{2},\varOmega } \le C({\widehat{\varOmega }},{\widehat{\varGamma }}_{\mathrm{D}}) \Vert \psi \Vert _{0,{\varGamma _{\mathrm{A}}}}. \end{aligned}$$

Then, (A.4) follows from standard approximation theory [34, 43].

On the other hand, when \(|{\varGamma _{\mathrm{D}}}| \ne 0\), we split \({\mathscr {L}}_\psi = \phi _2 + \phi _{\frac{3}{2}}\) where \(\phi _2, \phi _{\frac{3}{2}} \in H^1_{\varGamma _{\mathrm{D}}}(\varOmega )\) are uniquely defined by

$$\begin{aligned} ({\varvec{\nabla }}w,{\varvec{\nabla }}\phi _2 ) = -k^2 (w,{\mathscr {L}}_\psi ), \quad ({\varvec{\nabla }}w,{\varvec{\nabla }}\phi _{\frac{3}{2}}) = (w,\psi -ik{\mathscr {L}}_\psi )_{{\varGamma _{\mathrm{A}}}} \end{aligned}$$

for all \(w \in H^1_{\varGamma _{\mathrm{D}}}(\varOmega )\). Picking the test function \(w = \phi _2\), it follows that

$$\begin{aligned} |\phi _2|_{1,\varOmega }^2 \le k^2 \Vert {\mathscr {L}}_\psi \Vert _{0,\varOmega }\Vert \phi _2\Vert _{0,\varOmega } \le C({\widehat{\varOmega }},{\widehat{\varGamma }}_{\mathrm{D}}) k^{\frac{1}{2}} h_\varOmega \Vert \psi \Vert _{0,{\varGamma _{\mathrm{A}}}}|\phi _2|_{1,\varOmega }, \end{aligned}$$

where we used the Poincaré inequality to handle \(\Vert \phi _2\Vert _{0,\varOmega }\) and (A.3) to estimate \(k\Vert {\mathscr {L}}_\psi \Vert _{0,\varOmega }\). Similarly, employing a multiplicative trace inequality which combined with the Poincaré inequality yields \(\Vert \phi _{\frac{3}{2}}\Vert _{0,{\varGamma _{\mathrm{A}}}}\le C({\widehat{\varOmega }},{\widehat{\varGamma }}_{\mathrm{D}}) h_\varOmega ^{\frac{1}{2}} |\phi _{\frac{3}{2}}|_{1,\varOmega }\), we infer using (A.2) that

$$\begin{aligned} |\phi _{\frac{3}{2}}|_{1,\varOmega }^2 \le \Vert \psi - ik{\mathscr {L}}_\psi \Vert _{0,{\varGamma _{\mathrm{A}}}} \Vert \phi _{\frac{3}{2}}\Vert _{0,{\varGamma _{\mathrm{A}}}} \le C({\widehat{\varOmega }},{\widehat{\varGamma }}_{\mathrm{D}}) h_\varOmega ^{\frac{1}{2}} \Vert \psi \Vert _{0,{\varGamma _{\mathrm{A}}}}|\phi _{\frac{3}{2}}|_{1,\varOmega }. \end{aligned}$$

Then, invoking Assumption A.1, we have

$$\begin{aligned} |\phi _2|_{2,{\widetilde{\varOmega }}} \le C({\widehat{\varOmega }},{\widehat{\varGamma }}_{\mathrm{D}}) k^2 \Vert {\mathscr {L}}_\psi \Vert _{0,\varOmega } \le C({\widehat{\varOmega }},{\widehat{\varGamma }}_{\mathrm{D}}) k^{\frac{1}{2}} \Vert \psi \Vert _{0,{\varGamma _{\mathrm{A}}}}, \end{aligned}$$

and

$$\begin{aligned} |\phi _{\frac{3}{2}}|_{\frac{3}{2},{\widetilde{\varOmega }}} \le C({\widehat{\varOmega }},{\widehat{\varGamma }}_{\mathrm{D}}) \Vert \psi -ik{\mathscr {L}}_\psi \Vert _{0,{\varGamma _{\mathrm{A}}}} \le C({\widehat{\varOmega }},{\widehat{\varGamma }}_{\mathrm{D}}) \Vert \psi \Vert _{0,{\varGamma _{\mathrm{A}}}}. \end{aligned}$$

Thus, since \({\text {supp}}\chi \subset {\widetilde{\varOmega }}\) and invoking once again the Poincaré inequality, we infer that

$$\begin{aligned} |\chi \phi _2|_{2,\varOmega }&\le \Vert \chi \Vert _{0,\varOmega }|\phi _2|_{2,{\widetilde{\varOmega }}} + |\chi |_{1,\varOmega }|\phi _2|_{1,\varOmega } + |\chi |_{2,\varOmega }\Vert \phi _2\Vert _{0,\varOmega } \\&\le C({\widehat{\varOmega }},{\widehat{\varGamma }}_{\mathrm{D}}) \left( |\phi _2|_{2,{\widetilde{\varOmega }}} + h_\varOmega ^{-1} |\phi _2|_{1,\varOmega } \right) \\&\le C({\widehat{\varOmega }},{\widehat{\varGamma }}_{\mathrm{D}}) k^{\frac{1}{2}} \Vert \psi \Vert _{0,{\varGamma _{\mathrm{A}}}}, \end{aligned}$$

and similar arguments show that

$$\begin{aligned} |\chi \phi _{\frac{3}{2}}|_{\frac{3}{2},{\widetilde{\varOmega }}} \le C({\widehat{\varOmega }},{\widehat{\varGamma }}_{\mathrm{D}}) \Vert \psi \Vert _{0,{\varGamma _{\mathrm{A}}}}. \end{aligned}$$

At this point, (A.4) follows from standard approximation theory [34, 43]. \(\square \)

We are now ready to establish the main result of this appendix which proves the claim (A.6) below.

Proposition A.3

(Bound on \({{\widetilde{\sigma }}}_{\mathrm{ba}}\)) Let Assumption A.1 hold. Then

$$\begin{aligned} {{\widetilde{\sigma }}}_{\mathrm{ba}}\le C({\widehat{\varOmega }},{\widehat{\varGamma }}_{\mathrm{D}}) C_{\mathrm{qi}}(\kappa ) \left( \left( \frac{kh}{p} \right) ^{\frac{1}{2}} + \frac{kh}{p}\right) + \left( C({\widehat{\varOmega }},{\widehat{\varGamma }}_{\mathrm{D}}) \frac{1}{k h_\varOmega } \left( 1 + \frac{1}{k h_\varOmega }\right) + 2 \right) {\sigma _{\mathrm{ba}}}. \end{aligned}$$

Proof

We consider an arbitrary element \(\psi \in L^2({\varGamma _{\mathrm{A}}})\) and define \(u_\psi ^\star \) as the unique element of \(H^1_{\varGamma _{\mathrm{D}}}(\varOmega )\) such that \(b(w,u_\psi ^\star ) = (w,\psi )_{\varGamma _{\mathrm{A}}}\) for all \(w \in H^1_{\varGamma _{\mathrm{D}}}(\varOmega )\). Then, defining \({\mathscr {L}}_\psi \in H^1_{\varGamma _{\mathrm{D}}}(\varOmega )\) following Lemma A.2, we have

$$\begin{aligned}&b(w,\chi {\mathscr {L}}_\psi ) \\&\quad = -k^2 (w,\chi {\mathscr {L}}_\psi ) - ik (w,\chi {\mathscr {L}}_\psi )_{\varGamma _{\mathrm{A}}}+ ({\varvec{\nabla }}w, {\varvec{\nabla }}(\chi {\mathscr {L}}_\psi )) \\&\quad = k^2 (\chi w,{\mathscr {L}}_\psi ) - ik (\chi w,{\mathscr {L}}_\psi )_{\varGamma _{\mathrm{A}}}+ ({\varvec{\nabla }}(\chi w),{\varvec{\nabla }}{\mathscr {L}}_\psi ) \\&\qquad + ({\varvec{\nabla }}w, {\varvec{\nabla }}(\chi {\mathscr {L}}_\psi )) - ({\varvec{\nabla }}(\chi w),{\varvec{\nabla }}{\mathscr {L}}_\psi ) - 2k^2 (\chi w,{\mathscr {L}}_\psi ) \\&\quad = a(\chi w,{\mathscr {L}}_\psi ) + ({\varvec{\nabla }}w,{\mathscr {L}}_\psi {\varvec{\nabla }}\chi ) - (w{\varvec{\nabla }}\chi ,{\varvec{\nabla }}{\mathscr {L}}_\psi ) - 2k^2 (\chi w,{\mathscr {L}}_\psi ) \\&\quad = (\chi w,\psi )_{\varGamma _{\mathrm{A}}}+ ({\varvec{\nabla }}w,{\mathscr {L}}_\psi {\varvec{\nabla }}\chi ) - (w{\varvec{\nabla }}\chi ,{\varvec{\nabla }}{\mathscr {L}}_\psi ) - 2k^2 (\chi w,{\mathscr {L}}_\psi ) \\&\quad = (w,\psi )_{\varGamma _{\mathrm{A}}}+ (w,{\mathscr {L}}_\psi {\varvec{\nabla }}\chi {\cdot } {\varvec{n}})_{\partial \varOmega } - (w,{\varvec{\nabla }}{\cdot }({\mathscr {L}}_\psi {\varvec{\nabla }}\chi )) - (w,{\varvec{\nabla }}\chi {\cdot } {\varvec{\nabla }}{\mathscr {L}}_\psi ) - 2k^2 (w,\chi {\mathscr {L}}_\psi ) \\&\quad = (w,\psi )_{\varGamma _{\mathrm{A}}}- (w, {\varvec{\nabla }}{\cdot }({\mathscr {L}}_\psi {\varvec{\nabla }}\chi )+{\varvec{\nabla }}\chi {\cdot } {\varvec{\nabla }}{\mathscr {L}}_\psi + 2k^2 \chi {\mathscr {L}}_\psi ), \end{aligned}$$

where we used that \({\varvec{\nabla }}\chi {\cdot } {\varvec{n}}= 0\) on \({\varGamma _{\mathrm{A}}}\), as \(\chi = 1\) in a neighborhood of \({\varGamma _{\mathrm{A}}}\), and that \(w = 0\) on \({\varGamma _{\mathrm{D}}}\) so that \((w,{\mathscr {L}}_\psi {\varvec{\nabla }}\chi \cdot {\varvec{n}})_{\partial \varOmega } = 0\). It follows that

$$\begin{aligned} b(w,u_\psi ^\star - \chi {\mathscr {L}}_\psi ) = (w,{\widetilde{f}}), \end{aligned}$$

with \({\widetilde{f}} = {\varvec{\nabla }}{\cdot }({\mathscr {L}}_\psi {\varvec{\nabla }}\chi )+{\varvec{\nabla }}\chi {\cdot } {\varvec{\nabla }}{\mathscr {L}}_\psi +2k^2 \chi {\mathscr {L}}_\psi \). In particular, we have \({\widetilde{f}} \in L^2(\varOmega )\), and using (A.3), we infer that

$$\begin{aligned} \Vert {\widetilde{f}}\Vert _{0,\varOmega } \le \left( C({\widehat{\varOmega }},{\widehat{\varGamma }}_{\mathrm{D}}) \frac{1}{h_\varOmega } \left( 1 + \frac{1}{k h_\varOmega }\right) k^{-\frac{1}{2}} + 2k^{\frac{1}{2}} \right) \Vert \psi \Vert _{0,{\varGamma _{\mathrm{A}}}}. \end{aligned}$$
(A.5)

Then, we have

$$\begin{aligned} k^{\frac{1}{2}} \left( \inf _{v_h \in V_h} |u_\psi ^\star - v_h|_{1,\varOmega } \right)&\le k^{\frac{1}{2}} \left( \inf _{y_h \in V_h} |(u_\psi ^\star - \chi {\mathscr {L}}_\psi ) - y_h|_{1,\varOmega } + \inf _{w_h \in V_h} |\chi {\mathscr {L}}_\psi - w_h|_{1,\varOmega } \right) \\&\le k^{-\frac{1}{2}} {\sigma _{\mathrm{ba}}}\Vert {\widetilde{f}}\Vert _{0,\varOmega } + k^{\frac{1}{2}} \inf _{w_h \in V_h} |\chi {\mathscr {L}}_\psi - w_h|_{1,\varOmega }, \end{aligned}$$

where the first bound stems by taking the function \(v_h=y_h+w_h\) where \(y_h\in V_h\) and \(w_h\in V_h\) realize the two infimums in the right-hand side. Using (A.5) and Lemma A.2, we see that

$$\begin{aligned} k^{\frac{1}{2}} \left( \inf _{v_h \in V_h} |u_\psi ^\star - v_h|_{1,\varOmega } \right) \le {}&\left( C({\widehat{\varOmega }},{\widehat{\varGamma }}_{\mathrm{D}}) \frac{1}{k h_\varOmega } \left( 1 + \frac{1}{k h_\varOmega } \right) + 2 \right) {\sigma _{\mathrm{ba}}}\Vert \psi \Vert _{0,{\varGamma _{\mathrm{A}}}} \\&+ C({\widehat{\varOmega }},{\widehat{\varGamma }}_{\mathrm{D}}) C_{\mathrm{qi}}(\kappa ) \left( \frac{kh}{p} + \left( \frac{kh}{p} \right) ^{\frac{1}{2}} \right) \Vert \psi \Vert _{0,{\varGamma _{\mathrm{A}}}}, \end{aligned}$$

and we conclude by taking the supremum over \(\psi \in L^2({\varGamma _{\mathrm{A}}})\). \(\square \)

Corollary A.4

(Asymptotic regime) Under Assumption A.1, we have

$$\begin{aligned} 1 + {\tilde{\theta }}_2({\sigma _{\mathrm{ba}}},{{\widetilde{\sigma }}}_{\mathrm{ba}}) \le 1 + {\widetilde{\theta }}_3 \left( {\sigma _{\mathrm{ba}}},\frac{kh}{p}\right) , \end{aligned}$$
(A.6)

where \({\widetilde{\theta }}_3\) is a decreasing function of its two arguments such that \(\lim _{t,t' \rightarrow 0} {\widetilde{\theta }}_3(t,t') = 0\).

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Chaumont-Frelet, T., Ern, A. & Vohralík, M. On the derivation of guaranteed and p-robust a posteriori error estimates for the Helmholtz equation. Numer. Math. 148, 525–573 (2021). https://doi.org/10.1007/s00211-021-01192-w

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