High-dimensional finite elements for multiscale Maxwell-type equations

Abstract

We consider multiscale Maxwell-type equations in a domain |$D\subset\mathbb{R}^d$| (⁠|$d=2,3$|⁠), which depend on |$n$| microscopic scales. Using multiscale convergence, we derive the multiscale homogenized problem, which is posed in |$\mathbb{R}^{(n+1)d}$|⁠. 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 |$\mathbb{R}^d$|⁠. 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 |$H^1({\rm curl\,},D)$|⁠. However, in polygonal domains, |$u_0$| belongs only to a weaker regularity space |$H^s({\rm curl\,},D)$| 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 |$H^{1+s}(D)$| (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.