Decompositions of high-frequency Helmholtz solutions and application to the finite element method

Lafontaine, David

Séminaire Laurent Schwartz — EDP et applications (2021-2022), Exposé no. 16, 15 p.

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 $C^{\infty }$ variable coefficients in ${ℝ}^d$, 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 $hp$-finite-element method ($hp$-FEM) applied to the $C^{\infty }$ variable-coefficient Helmholtz equation in ${ℝ}^d$, (ii) the $hp$-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 $h$-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].

[1] I. M. Babuška and S. A. Sauter. Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave numbers? SIAM Review, pages 451–484, 2000. | Zbl

[2] T. Betcke, S. N. Chandler-Wilde, I. G. Graham, S. Langdon, and M. Lindner. Condition number estimates for combined potential boundary integral operators in acoustics and their boundary element discretisation. Numer. Methods Partial Differential Eq., 27(1):31–69, 2011. | DOI | MR | Zbl

[3] N. Burq. Semi-classical estimates for the resolvent in nontrapping geometries. International Mathematics Research Notices, 2002(5):221–241, 2002. | DOI | Zbl

[4] F. Cardoso and G. Popov. Quasimodes with exponentially small errors associated with elliptic periodic rays. Asymptotic Analysis, 30(3, 4):217–247, 2002. | Zbl

[5] M. Costabel, M. Dauge, and S. Nicaise. Corner Singularities and Analytic Regularity for Linear Elliptic Systems. Part I: Smooth domains. 2010. | HAL

[6] E. B. Davies. The functional calculus. J. London Math. Soc. (2), 52(1):166–176, 1995. | DOI | MR | Zbl

[7] E. B. Davies. Spectral theory and differential operators. Cambridge University Press, 1995. | DOI | Zbl

[8] S. Dyatlov and M. Zworski. Mathematical theory of scattering resonances, volume 200 of Graduate Studies in Mathematics. American Mathematical Society, 2019. | DOI | Zbl

[9] L. Escauriaza, S. Montaner, and C. Zhang. Analyticity of solutions to parabolic evolutions and applications. SIAM J. Math. Anal., 49(5):4064–4092, 2017. | DOI | MR | Zbl

[10] S. Esterhazy and J. M. Melenk. On stability of discretizations of the Helmholtz equation. In I. G. Graham, T. Y. Hou, O. Lakkis, and R. Scheichl, editors, Numerical Analysis of Multiscale Problems, pages 285–324. Springer, 2012. | DOI | Zbl

[11] A. Friedman. Partial differential equations. Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London, 1969.

[12] J. Galkowski, D. Lafontaine, E. A. Spence, and J. Wunsch. Decompositions of high-frequency Helmholtz solutions via functional calculus, and application to the finite element method. Submitted. | arXiv

[13] J. Galkowski, E. A. Spence, and J. Wunsch. Optimal constants in nontrapping resolvent estimates. Pure and Applied Analysis, 2(1):157–202, 2020. | DOI | MR | Zbl

[14] B. Helffer and J. Sjöstrand. Équation de Schrödinger avec champ magnétique et équation de Harper. In Schrödinger operators (Sønderborg, 1988), volume 345 of Lecture Notes in Phys., pages 118–197. Springer, Berlin, 1989. | DOI | Zbl

[15] D. Lafontaine, E. A. Spence, and J. Wunsch. For most frequencies, strong trapping has a weak effect in frequency-domain scattering. Communications on Pure and Applied Mathematics, 74(10):2025–2063, 2021. | DOI | MR | Zbl

[16] D. Lafontaine, E. A. Spence, and J. Wunsch. Wavenumber-explicit convergence of the hp-FEM for the full-space heterogeneous Helmholtz equation with smooth coefficients. Comp. Math. Appl., 113:59–69, 2022. | DOI | MR | Zbl

[17] P. D. Lax and R. S. Phillips. Scattering Theory. Academic Press, revised edition, 1989. | Zbl

[18] W. McLean. Strongly elliptic systems and boundary integral equations. Cambridge University Press, 2000. | Zbl

[19] J. M. Melenk, A. Parsania, and S. Sauter. General DG-methods for highly indefinite Helmholtz problems. Journal of Scientific Computing, 57(3):536–581, 2013. | DOI | MR | Zbl

[20] J. M. Melenk and S. Sauter. Convergence analysis for finite element discretizations of the Helmholtz equation with Dirichlet-to-Neumann boundary conditions. Math. Comp, 79(272):1871–1914, 2010. | DOI | MR | Zbl

[21] J. M. Melenk and S. Sauter. Wavenumber explicit convergence analysis for Galerkin discretizations of the Helmholtz equation. SIAM J. Numer. Anal., 49:1210–1243, 2011. | DOI | MR | Zbl

[22] R. B. Melrose and J. Sjöstrand. Singularities of boundary value problems. II. Communications on Pure and Applied Mathematics, 35(2):129–168, 1982. | DOI | MR | Zbl

[23] G. S. Popov. Quasimodes for the Laplace operator and glancing hypersurfaces. In Microlocal Analysis and Nonlinear Waves, pages 167–178. Springer, 1991. | DOI | Zbl

[24] M. Reed and B. Simon. Methods of Modern Mathematical Physics Volume 1: Functional analysis. New York-London: Academic Press, Inc, 1972. | Zbl

[25] J. Sjöstrand. A trace formula and review of some estimates for resonances. In Microlocal analysis and spectral theory (Lucca, 1996), volume 490 of NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., pages 377–437. Kluwer Acad. Publ., Dordrecht, 1997. | DOI | Zbl

[26] J. Sjöstrand and M. Zworski. Complex scaling and the distribution of scattering poles. J. Amer. Math. Soc., 4(4):729–769, 1991. | DOI | MR | Zbl

[27] M. Zworski. Semiclassical analysis, volume 138 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2012. | DOI | Zbl