Edge multiscale finite element methods for semilinear parabolic problems with heterogeneous coefficients

TL;DR

This paper introduces a new edge multiscale finite element method for efficiently solving semilinear parabolic problems with heterogeneous coefficients, achieving accuracy independent of heterogeneity through a specialized multiscale ansatz space.

Contribution

The paper develops a novel spatial semidiscrete multiscale method based on edge multiscale techniques, with a new ansatz space construction for semilinear parabolic problems with heterogeneous coefficients.

Findings

Method achieves accuracy independent of heterogeneity.

Error estimates depend on coarse grid size and level parameter.

Numerical experiments confirm efficiency and accuracy.

Abstract

We develop a new spatial semidiscrete multiscale method based upon the edge multiscale methods to solve semilinear parabolic problems with heterogeneous coefficients and smooth initial data. This method allows for a cheap spatial discretization, which fails to resolve the spatial heterogeneity but maintains satisfactory accuracy independent of the heterogeneity. This is achieved by simultaneously constructing a steady-state multiscale ansatz space with certain approximation properties for the evolving solution and the initial data. The approximation properties of the multiscale ansatz space are derived using local-global splitting. A fully discrete scheme is analyzed using a first-order explicit exponential Euler scheme. We derive the error estimates in the $L^{2}$-norm and energy norm under the regularity assumptions for the semilinear term. The convergence rates depend on the coarse grid size and the level parameter. Finally, extensive numerical experiments are carried out to validate the efficiency of the proposed method.