2016/11/03 by Erik Eikeland, Eikeland, Erik, Leszek Marcinkowski +3
Computer Science · Engineering · #65F08 (Secondary) #65N30 #65N55 (Primary) #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #Composite Material Mechanics #FOS: Mathematics #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.1611.00968
openalex publication_date 2016/11/03 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28
We propose two variants of the overlapping additive Schwarz method for the\nfinite element discretiza- tion of the elliptic problem in 3D with highly\nheterogeneous coefficients. The methods are efficient and simple to construct\nusing the abstract framework of the additive Schwarz method, and an idea of\nadaptive coarse spaces. In one variant, the coarse space consists of finite\nelement functions associated with the wire basket nodes and functions based on\nsolving some generalized eigenvalue problem on the faces, and in the other\nvariant, it contains functions associated with the vertex nodes with functions\nbased on solving some generalized eigenvalue problems on subdomain faces and on\nsubdomain edges. The functions that are used to build the coarse spaces are\nchosen adaptively, they correspond to the eigenvalues that are smaller than a\ngiven threshold. The convergence rate of the preconditioned conjugate gradients\nmethod in both cases, is shown to be independent of the variations in the\ncoefficients for sufficient number of eigenfunctions in the coarse space.\nNumerical results are given to support the theory.\n