Sparse Finite Element Method for Periodic Multiscale Nonlinear Monotone Problems

A sparse tensor product finite element (FE) method is developed for the high-dimensional limiting problem obtained by applying the multiscale convergence to a multiscale nonlinear monotone problem in $\mathbb{R}^d$. The limiting problem is posed in a product space, so tensor product FE spaces are used for discretization. This sparse FE method requires essentially the same number of degrees of freedom to achieve essentially equal accuracy to that of a standard FE scheme for a partial differential equation in $\mathbb{R}^d$. It is proved that for the Euler–Lagrange equation of a two scale convex variational problem in a smooth and convex domain the solution to the high-dimensional limiting equation is smooth. An analytic homogenization error is then established, which together with the FE error provides an explicit error estimate for an approximation to the solution of the original multiscale problem. Without this regularity, such an approximation always exists when the meshsize and the microscale converge ...