Wavenumber explicit convergence of a multiscale generalized finite element method for heterogeneous Helmholtz problems

• Verfasst von: Chupeng, Ma [VerfasserIn]
• Verfasst von: Alber, Christian [VerfasserIn]
• Verfasst von: Scheichl, Robert [VerfasserIn]
• Titel: Wavenumber explicit convergence of a multiscale generalized finite element method for heterogeneous Helmholtz problems
• Verf.angabe: Ma Chupeng, Christian Alber, and Robert Scheichl
• E-Jahr: 2023
• Jahr: Jun 2023
• Umfang: 39 S.
• Fussnoten: Online veröffentlicht: 15. Juni 2023 ; Gesehen am 26.09.2023
• Titel Quelle: Enthalten in: Society for Industrial and Applied MathematicsSIAM journal on numerical analysis
• Ort Quelle: Philadelphia, Pa. : SIAM, 1966
• Jahr Quelle: 2023
• Band/Heft Quelle: 61(2023), 3 vom: Juni, Seite 1546-1584
• ISSN Quelle: 1095-7170
• DOI: doi:10.1137/21M1466748
• Datenträger: Online-Ressource
• Sprache: eng
• K10plus-PPN: 186027823X

Abstract

In this paper, the generalized finite element method (GFEM) for solving second order elliptic equations with rough coefficients is studied. New optimal local approximation spaces for GFEMs based on local eigenvalue problems involving a partition of unity are presented. These new spaces have advantages over those proposed in [I. Babuska and R. Lipton, Multiscale Model. Simul., 9 (2011), pp. 373--406]. First, in addition to a nearly exponential decay rate of the local approximation errors with respect to the dimensions of the local spaces, the rate of convergence with respect to the size of the oversampling region is also established. Second, the theoretical results hold for problems with mixed boundary conditions defined on general Lipschitz domains. Finally, an efficient and easy-to-implement technique for generating the discrete $A$-harmonic spaces is proposed which relies on solving an eigenvalue problem associated with the Dirichlet-to-Neumann operator, leading to a substantial reduction in computational cost. Numerical experiments are presented to support the theoretical analysis and to confirm the effectiveness of the new method.