Two-scale Finite Element Discretizations for Semilinear Parabolic Equations

In this paper, to reduce the computational cost of solving semilinear parabolic equations on a tensor product domain Ω⊂ℝd with d = 2 or 3, some two-scale finite element discretizations are proposed and analyzed. The time derivative in semilinear parabolic equations is approximated by the backward Euler finite difference scheme. The two-scale finite element method is designed for the space discretization. The idea of the two-scale finite element method is based on an understanding of a finite element solution to an elliptic problem on a tensor product domain. The high frequency parts of the finite element solution can be well captured on some univariate fine grids and the low frequency parts can be approximated on a coarse grid. Thus the two-scale finite element approximation is defined as a linear combination of some standard finite element approximations on some univariate fine grids and a coarse grid satisfying H = O (h1/2), where h and H are the fine and coarse mesh widths, respectively. It is shown theoretically and numerically that the backward Euler two-scale finite element solution not only achieves the same order of accuracy in the H1 (Ω) norm as the backward Euler standard finite element solution, but also reduces the number of degrees of freedom from O(h-d×τ-1) to O(h-((d)+1)/2×τ-1) where τ is the time step. Consequently the backward Euler two-scale finite element method for semilinear parabolic equations is more efficient than the backward Euler standard finite element method.