Error Estimates for Two-Scale Composite Finite Element Approximations of Nonlinear Parabolic Equations

Abstract

We study spatially semidiscrete and fully discrete two-scale composite finite element method for approximations of the nonlinear parabolic equations with homogeneous Dirichlet boundary conditions in a convex polygonal domain in the plane. This new class of finite elements, which is called composite finite elements, was first introduced by Hackbusch and Sauter [Numer. Math., 75 (1997), pp. 447-472] for the approximation of partial differential equations on domains with complicated geometry. The aim of this paper is to introduce an efficient numerical method which gives a lower dimensional approach for solving partial differential equations by domain discretization method. The composite finite element method introduces two-scale grid for discretization of the domain, the coarse-scale and the fine-scale grid with the degrees of freedom lies on the coarse-scale grid only. While the fine-scale grid is used to resolve the Dirichlet boundary condition, the dimension of the finite element space depends only on the coarse-scale grid. As a consequence, the resulting linear system will have a fewer number of unknowns. A continuous, piecewise linear composite finite element space is employed for the space discretization whereas the time discretization is based on both the backward Euler and the Crank-Nicolson methods. We have derived the error estimates in the $L^\infty(L^2)$-norm for both semidiscrete and fully discrete schemes. Moreover, numerical simulations show that the proposed method is an efficient method to provide a good approximate solution.