Novel design and analysis of generalized finite element methods based on locally optimal spectral approximations

• Verfasst von: Ma, Chupeng [VerfasserIn]
• Verfasst von: Scheichl, Robert [VerfasserIn]
• Verfasst von: Dodwell, Tim [VerfasserIn]
• Titel: Novel design and analysis of generalized finite element methods based on locally optimal spectral approximations
• Verf.angabe: Chupeng Ma, Robert Scheichl, and Tim Dodwell
• E-Jahr: 2022
• Jahr: January 31, 2022
• Umfang: 30 S.
• Fussnoten: Gesehen am 22.11.2022
• Titel Quelle: Enthalten in: Society for Industrial and Applied MathematicsSIAM journal on numerical analysis
• Ort Quelle: Philadelphia, Pa. : SIAM, 1966
• Jahr Quelle: 2022
• Band/Heft Quelle: 60(2022), 1, Seite 244-273
• ISSN Quelle: 1095-7170
• DOI: doi:10.1137/21M1406179
• Datenträger: Online-Ressource
• Sprache: eng
• Sach-SW: elliptic problems
• Sach-SW: generalized finite element method
• Sach-SW: homogenization
• Sach-SW: Kolo-mogrov n-width
• Sach-SW: local spectral basis
• Sach-SW: multiscale
• Sach-SW: multiscale method
• Sach-SW: partition of unity
• Sach-SW: spaces
• K10plus-PPN: 1823165435

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, Multisvale 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.