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 High-dimensional finite elements for multiscale Maxwell-type equations
Tác giả hoặc Nhóm tác giả: Chu Van Tiep, Hoang Viet Ha
Nơi đăng: IMA Journal of Numerical Analysis; Số: 38 (1);Từ->đến trang: 227-270;Năm: 2018
Lĩnh vực: Khoa học công nghệ; Loại: Bài báo khoa học; Thể loại: Quốc tế
TÓM TẮT
We consider multiscale Maxwell-type equations in a domain (), which depend on microscopic scales. Using multiscale convergence, we derive the multiscale homogenized problem, which is posed in . Solving it, we get all the necessary macroscopic and microscopic information. Sparse tensor product finite elements (FEs) are employed, using edge FEs. The method achieves a required level of accuracy with essentially an optimal number of degrees of freedom, which, apart from a multiplying logarithmic term, is equal to that for solving a problem in . Numerical correctors are constructed from the FE solutions. In the two-scale case, an explicit homogenization error is deduced. To get this error, the standard procedure in the homogenization literature requires the solution of the homogenized problem to belong to . However, in polygonal domains, belongs only to a weaker regularity space
ABSTRACT
We consider multiscale Maxwell-type equations in a domain (), which depend on microscopic scales. Using multiscale convergence, we derive the multiscale homogenized problem, which is posed in . Solving it, we get all the necessary macroscopic and microscopic information. Sparse tensor product finite elements (FEs) are employed, using edge FEs. The method achieves a required level of accuracy with essentially an optimal number of degrees of freedom, which, apart from a multiplying logarithmic term, is equal to that for solving a problem in . Numerical correctors are constructed from the FE solutions. In the two-scale case, an explicit homogenization error is deduced. To get this error, the standard procedure in the homogenization literature requires the solution of the homogenized problem to belong to . However, in polygonal domains, belongs only to a weaker regularity space
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