Sparse tensor product finite element method for nonlinear multiscale variational inequalities of monotone type

Abstract

Abstract We study an essentially optimal finite element (FE) method for locally periodic nonlinear multiscale variational inequalities of monotone type in a domain $D\subset{\mathbb{R}}^d$ that depend on a macroscopic and $n$ microscopic scales. The scales are separable. Using multiscale convergence we deduce a multiscale homogenized variational inequality in a tensorized domain in the high-dimensional space ${\mathbb R}^{(n+1)d}$. Given sufficient regularity on the solution the sparse tensor product FE method is developed for this problem, which attains an essentially equal (i.e., it differs by only a logarithmic factor) level of accuracy to that of the full tensor product FE method, but requires an essentially optimal number of degrees of freedom which is equal to that for solving a problem in ${{\mathbb{R}}}^d$ apart from a logarithmic factor. For two-scale problems we deduce a new homogenization error for the nonlinear monotone variational inequality. A numerical corrector is then constructed with an explicit error in terms of the homogenization and the FE errors. For general multiscale problems we deduce a numerical corrector from the FE solution of the multiscale homogenized problem, but without an explicit error as such a homogenization error is not available.