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Title: High dimensional finite elements for two-scale Maxwell wave equations
Authors: Chu, Van Tiep
Hoang, Viet Ha
Keywords: Science::Mathematics
Issue Date: 2020
Source: Chu, V. T. & Hoang, V. H. (2020). High dimensional finite elements for two-scale Maxwell wave equations. Journal of Computational and Applied Mathematics, 375, 112756-.
Project: MOE2017-T2-2-144
Journal: Journal of Computational and Applied Mathematics
Abstract: We develop an essentially optimal numerical method for solving two-scale Maxwell wave equations in a domain D⊂Rd. 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 Rd, 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 Rd 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.
ISSN: 0377-0427
DOI: 10.1016/
Schools: School of Physical and Mathematical Sciences 
Rights: © 2020 Elsevier B.V. All rights reserved.
Fulltext Permission: none
Fulltext Availability: No Fulltext
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