High dimensional finite elements for two-scale Maxwell wave equations
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 microscop...
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| Main Authors: | , |
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| Other Authors: | |
| Format: | Article |
| Language: | English |
| Published: |
2022
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| Subjects: | |
| Online Access: | https://hdl.handle.net/10356/159512 |
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| Institution: | Nanyang Technological University |
| Language: | English |
| Summary: | 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. |
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