This paper presents joint works with Jeffrey Galkowski, Euan Spence, and Jared Wunsch [16, 12]. It corresponds to the talk the author gave at IHES for the Séminaire Laurent Schwartz in April 2022.
Over the last ten years, results of Melenk and Sauter [20, 21] decomposing high-frequency Helmholtz solutions into an analytic part and a well-behaved in frequency part have had a large impact in the numerical analysis of the Helmholtz equation. These results have been proved for the constant-coefficients Helmholtz equation outside an analytic Dirichlet obstacle or an interior domain with an impedance boundary condition.
In [16], we obtained an analogous decomposition for the Helmholtz equation with variable coefficients in , then in [12], analogous decompositions for scattering problems fitting into the very general black-box scattering framework of Sjöstrand and Zworski [26], thus covering Helmholtz problems with variable coefficients, impenetrable obstacles, and penetrable obstacles all at once. These results allowed us to prove new sharp frequency-explicit convergence results for (i) the -finite-element method (-FEM) applied to the variable-coefficient Helmholtz equation in , (ii) the -FEM applied to the variable-coefficient Helmholtz equation in the exterior of an analytic Dirichlet obstacle, where the coefficients are analytic in a neighborhood of the obstacle, and (iii) the -FEM applied to the Helmholtz penetrable-obstacle transmission problem. In this expository paper, we show how to obtain the decomposition from [16], and the main ideas behind the general result of [12].
@article{SLSEDP_2021-2022____A7_0, author = {David Lafontaine}, title = {Decompositions of high-frequency {Helmholtz} solutions and application to the finite element method}, journal = {S\'eminaire Laurent Schwartz {\textemdash} EDP et applications}, note = {talk:16}, pages = {1--15}, publisher = {Institut des hautes \'etudes scientifiques & Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique}, year = {2021-2022}, doi = {10.5802/slsedp.152}, language = {en}, url = {https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.152/} }
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%0 Journal Article %A David Lafontaine %T Decompositions of high-frequency Helmholtz solutions and application to the finite element method %J Séminaire Laurent Schwartz — EDP et applications %Z talk:16 %D 2021-2022 %P 1-15 %I Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique %U https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.152/ %R 10.5802/slsedp.152 %G en %F SLSEDP_2021-2022____A7_0
David Lafontaine. Decompositions of high-frequency Helmholtz solutions and application to the finite element method. Séminaire Laurent Schwartz — EDP et applications (2021-2022), Talk no. 16, 15 p. doi : 10.5802/slsedp.152. https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.152/
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