High dimensional finite elements for two-scale Maxwell wave equations

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS

卷:375

Article

Van Tiep Chu^1,2  Viet Ha Hoang¹ 

[1] Nanyang Technol Univ, Sch Phys & Math Sci, Div Math Sci, Singapore 637371, Singapore

[2] Univ Danang, Dept Math, Univ Sci & Educ, 459 Ton Duc Thang, Danang, Vietnam

关键词: Multiscale Maxwell wave equation;    Finite elements;    High dimension;    Optimal complexity;    Numerical corrector;   

DOI  :  10.1016/j.cam.2020.112756

来源: Elsevier

【 摘 要 】

We develop an essentially optimal numerical method for solving two-scale Maxwell wave equations in a domain D subset of R-d. The problems depend on two scales: one macroscopic scale and one microscopic scale. Solving the macroscopic two-scale homogenized problem, we obtain the desired macroscopic and microscopic information. This problem depends on two variables in R-d, one for each scale that the original two-scale equation depends on, and is thus posed in a high dimensional tensorized domain. The straightforward full tensor product finite element (FE) method is exceedingly expensive. We develop the sparse tensor product FEs that solve this two-scale homogenized problem with essentially optimal number of degrees of freedom, i.e. the number of degrees of freedom differs by only a logarithmic multiplying factor from that required for solving a macroscopic problem in a domain in R-d only, for obtaining a required level of accuracy. Numerical correctors are constructed from the FE solution. We derive a rate of convergence for the numerical corrector in terms of the microscopic scale and the FE mesh width. Numerical examples confirm our analysis. (C) 2020 Elsevier B.V. All rights reserved.

【 授权许可】

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