Sparse tensor product finite elements for two scale elliptic and parabolic equations with discontinuous coefficients

The paper develops the essentially optimal sparse tensor product finite element method for solving two scale elliptic and parabolic problems in a domain $D\subset R^d$, $d=2,3$, which is embedded with a periodic array of inclusions of microscopic sizes and spacing. The two scale coefficient is thus discontinuous in the fast variable. We obtain approximations for the solution of the homogenized equation and the scale interaction term, i.e. all the macroscopic and microscopic information, within a prescribed level of accuracy, using only an essentially optimal number of degrees of freedom, which is equal (apart from a possible logarithmic factor) to that required to solve one macroscopic scale problem in D. This is achieved by solving the two scale homogenized problem, and utilizing the regularity of the scale interaction term in all the slow and fast variables at the same time. However, unlike problems considered in the literature (e.g. Hoang and Schwab, 2004/05 [16]), the scale interaction term is only piecewise regular in the fast variable. We employ the discretization schemes developed for interface problems (Chen and Zou, 1998 [6], and Li et al., 2010 [20]) for the fast variable. Numerical correctors are developed from the finite element solutions with errors in terms of the finite element mesh size and the microscopic scale. Numerical examples that verify the theoretical convergence rates of the sparse tensor product finite elements are presented.