High-dimensional finite elements for multiscale Maxwell-type equations

Abstract

We consider multiscale Maxwell-type equations in a domain D C Rd (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 Rd . 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 solution u0 of the homogenized problem to belong to H1. However, in polygonal domains, u0 belongs only to a weaker regularity space Hs for 0 less than s less than 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 H2(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.