Two-scale Finite Element Discretizations for Semilinear Parabolic Equations

Abstract

In this paper, to reduce the computational cost of solving semilinear parabolic equations on a tensor product domain Ω⊂ℝ^<i>d</i> with <i>d</i> = 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 <i>H</i> = <i>O</i> (<i>h</i>^1/2), where <i>h</i> and <i>H</i> 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 <i>H</i>¹ (Ω) norm as the backward Euler standard finite element solution, but also reduces the number of degrees of freedom from <i>O</i>(<i>h</i>^-<i>d</i>×<i>τ</i>^-1) to <i>O</i>(<i>h</i>^{-(<i>(d)</i>+1)/2}×<i>τ</i>^-1) where <i>τ</i> 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.