Sparse tensor product high dimensional finite elements for two-scale mixed problems

Abstract

We develop the essentially optimal sparse tensor product finite element method for solving two-scale mixed problems in both the primal and dual forms. We study the two-scale homogenized mixed problems which are obtained in the limit where the microscopic scale tends to zero. These limiting problems are posed in a high dimensional tensorized product domain. Using sparse tensor product finite elements, we solve these problems with essentially optimal complexity to obtain an approximation for the solution within a prescribed accuracy. This is achieved when the solutions of the high dimensional two-scale homogenized mixed problems possess sufficient regularity with respect to both the slow and the fast variable at the same time. We show that this regularity requirement holds when the two-scale coefficient and the forcing satisfy mild smoothness conditions. From the finite element solutions for the two-scale homogenized mixed problems, we construct numerical correctors for the solutions of the original two-scale mixed problems. We prove an error estimate for these correctors in terms of both the homogenization error and the finite element error. Numerical examples confirm the theoretical results.