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Exponential Convergence for Multiscale Linear Elliptic PDEs via Adaptive Edge Basis Functions

Chen, Yifan and Hou, Thomas Y. and Wang, Yixuan (2021) Exponential Convergence for Multiscale Linear Elliptic PDEs via Adaptive Edge Basis Functions. Multiscale Modeling and Simulation, 19 (2). pp. 980-1010. ISSN 1540-3459. doi:10.1137/20m1352922.

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In this paper, we introduce a multiscale framework based on adaptive edge basis functions to solve second-order linear elliptic PDEs with rough coefficients. One of the main results is that we prove that the proposed multiscale method achieves nearly exponential convergence in the approximation error with respect to the computational degrees of freedom. Our strategy is to perform an energy orthogonal decomposition of the solution space into a coarse scale component comprising a-harmonic functions in each element of the mesh, and a fine scale component named the bubble part that can be computed locally and efficiently. The coarse scale component depends entirely on function values on edges. Our approximation on each edge is made in the Lions--Magenes space H_₀₀^(1/2)(e), which we will demonstrate to be a natural and powerful choice. We construct edge basis functions using local oversampling and singular value decomposition. When local information of the right-hand side is adaptively incorporated into the edge basis functions, we prove a nearly exponential convergence rate of the approximation error. Numerical experiments validate and extend our theoretical analysis; in particular, we observe no obvious degradation in accuracy for high-contrast media problems.

Item Type:Article
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URLURL TypeDescription Paper
Wang, Yixuan0000-0001-7305-5422
Additional Information:© 2021 Society for Industrial and Applied Mathematics. Received by the editors July 14, 2020; accepted for publication (in revised form) April 14, 2021; published electronically June 8, 2021. This research was supported in part by NSF grants DMS-1912654 and DMS-1907977. The first author was supported by the Caltech Kortchak Scholar Program.
Funding AgencyGrant Number
Caltech Kortchak Scholar ProgramUNSPECIFIED
Subject Keywords:multiscale PDEs, energy orthogonal decomposition, edge basis function, adaptive method, oversampling, exponential convergence
Issue or Number:2
Classification Code:AMS subject classifications. 65N30, 35J25, 65N15, 31A35
Record Number:CaltechAUTHORS:20210922-170252834
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Official Citation:Exponential Convergence for Multiscale Linear Elliptic PDEs via Adaptive Edge Basis Functions. Yifan Chen, Thomas Y. Hou, and Yixuan Wang. Multiscale Modeling & Simulation 2021 19:2, 980-1010; DOI: 10.1137/20m1352922
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:110985
Deposited By: Tony Diaz
Deposited On:22 Sep 2021 18:41
Last Modified:22 Sep 2021 18:41

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