A two-level overlapping Schwarz method with energy-minimizing multiscale coarse basis functions

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS

卷:370

Article

Wang, Junxian¹  Chung, Eric²  Kim, Hyea Hyun^3,4 

[1] Xiangtan Univ, Sch Math & Computat Sci, Hunan Key Lab Computat & Simulat Sci & Engn, Xiangtan 411105, Hunan, Peoples R China

[2] Chinese Univ Hong Kong, Dept Math, Hong Kong, Peoples R China

[3] Kyung Hee Univ, Dept Appl Math, Seoul, South Korea

[4] Kyung Hee Univ, Inst Nat Sci, Seoul, South Korea

关键词: Overlapping Schwarz method;    High contrast;    Multiscale finite element basis;    Coarse problem;   

DOI  :  10.1016/j.cam.2019.112600

来源: Elsevier

【 摘 要 】

A two-level overlapping Schwarz method is developed for second order elliptic problems with highly oscillatory and high contrast coefficients, for which it is known that the standard coarse problem fails to give a robust preconditioner. In this paper, we develop energy minimizing multiscale finite element functions to form a more robust coarse problem. First, a local spectral problem is solved in each non-overlapping coarse subdomain, and dominant eigenfunctions are selected as auxiliary functions, which are crucial for the high contrast case. The required multiscale basis functions are then obtained by minimizing an energy subject to some orthogonality conditions with respect to the auxiliary functions. Due to an exponential decay property, the minimization problem is solved locally on oversampling subdomains, that are unions of a few coarse subdomains. The coarse basis functions are therefore local and can be computed efficiently. The resulting preconditioner is shown to be robust with respect to the contrast in the coefficients as well as the overlapping width in the subdomain partition. Numerical results are included to confirm the theory and show the performance. (C) 2019 Elsevier B.V. All rights reserved.

【 授权许可】

Free