High-dimensional finite elements for multiscale Maxwell-type equations

We consider multiscale Maxwell-type equations in a domain D subset of R-d (d = 2, 3), which depend on n microscopic scales. Using multiscale convergence, we derive the multiscale homogenized problem, which is posed in R(n+1)d. 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 R-d. 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 u(0) of the homogenized problem to belong to H-1(cur1, D). However, in polygonal domains, u(0) belongs only to a weaker regularity space H-s(curl, D) for 0 < s < 1. We derive a homogenization error estimate for this case. Though we prove the result for two-scale Maxwell-type equations, the approach works verbatim for elliptic and elasticity problems when the solution to the homogenized equation belongs to H1+s(D) (standard procedure requires H-2(D) regularity). This homogenization error estimate is new in the literature. Thus, for two-scale problems, an explicit error for the numerical corrector is obtained; it is of the order of the sum of the homogenization error and the FE error. For the case of more than two scales, we construct a numerical corrector, albeit without a rate of convergence, as such a homogenization error is not available. Numerical experiments confirm the theoretical results.