<p>In this paper, we recast the variational formulation corresponding to the single layer boundary integral operator <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\varvec{{\text {\textsf{V}}}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext mathvariant="sans-serif">V</mtext> </mrow> </math></EquationSource> </InlineEquation> for the wave equation as a minimization problem in <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\varvec{L^2(\Sigma )}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi mathvariant="bold-italic">L</mi> <mn mathvariant="bold">2</mn> </msup> <mrow> <mo mathvariant="bold" stretchy="false">(</mo> <mi mathvariant="bold">Σ</mi> <mo mathvariant="bold" stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\varvec{\Sigma := \partial \Omega \times (0,T)}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold">Σ</mi> <mo mathvariant="bold">:</mo> <mo mathvariant="bold">=</mo> <mi mathvariant="bold-italic">∂</mi> <mi mathvariant="bold">Ω</mi> <mo mathvariant="bold">×</mo> <mo mathvariant="bold" stretchy="false">(</mo> <mn mathvariant="bold">0</mn> <mo mathvariant="bold">,</mo> <mi mathvariant="bold-italic">T</mi> <mo mathvariant="bold" stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is the lateral boundary of the space-time domain <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\varvec{Q:= \Omega \times (0,T)}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">Q</mi> <mo mathvariant="bold">:</mo> <mo mathvariant="bold">=</mo> <mi mathvariant="bold">Ω</mi> <mo mathvariant="bold">×</mo> <mo mathvariant="bold" stretchy="false">(</mo> <mn mathvariant="bold">0</mn> <mo mathvariant="bold">,</mo> <mi mathvariant="bold-italic">T</mi> <mo mathvariant="bold" stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. For discretization, the minimization problem is restated as a mixed saddle point formulation. Unique solvability is established by combining conforming nested boundary element spaces for the mixed formulation such that the related bilinear form is discrete inf-sup stable. We analyze under which conditions the discrete inf-sup stability is satisfied, and moreover, we show that the mixed formulation provides a simple error indicator, which can be used for adaptivity. We present several numerical experiments showing the applicability of the method to different time-domain boundary integral formulations used in the literature.</p>

Stable least-squares space-time boundary element methods for the wave equation

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2026年1月27日

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Daniel Hoonhout,
Richard Löscher,
Olaf Steinbach,
Carolina Urzúa–Torres

摘要

In this paper, we recast the variational formulation corresponding to the single layer boundary integral operator $\varvec{{\text {\textsf{V}}}}$ $V$ for the wave equation as a minimization problem in $\varvec{L^2(\Sigma )}$ $L^{2} (Σ)$ , where $\varvec{\Sigma := \partial \Omega \times (0,T)}$ $Σ : = \partial Ω \times (0, T)$ is the lateral boundary of the space-time domain $\varvec{Q:= \Omega \times (0,T)}$ $Q : = Ω \times (0, T)$ . For discretization, the minimization problem is restated as a mixed saddle point formulation. Unique solvability is established by combining conforming nested boundary element spaces for the mixed formulation such that the related bilinear form is discrete inf-sup stable. We analyze under which conditions the discrete inf-sup stability is satisfied, and moreover, we show that the mixed formulation provides a simple error indicator, which can be used for adaptivity. We present several numerical experiments showing the applicability of the method to different time-domain boundary integral formulations used in the literature.