Constraint energy minimizing generalized multiscale finite element method for convection diffusion equation

TL;DR

This paper introduces a constraint energy minimizing generalized multiscale finite element method for convection diffusion equations, providing a robust approach with proven convergence and good decay properties even in high contrast scenarios.

Contribution

The paper develops a novel multiscale finite element method using constraint energy minimization and oversampling, improving accuracy and convergence for convection diffusion problems with heterogeneous coefficients.

Findings

Method achieves convergence rate proportional to coarse mesh size.

Multiscale basis functions exhibit good decay properties.

Performance demonstrated through numerical experiments.

Abstract

In this paper we present and analyze a constraint energy minimizing generalized multiscale finite element method for convection diffusion equation. To define the multiscale basis functions, we first build an auxiliary multiscale space by solving local spectral problems motivated by analysis. Then constraint energy minimization performed in oversampling domains is exploited to construct the multiscale space. The resulting multiscale basis functions have a good decay property even for high contrast diffusion and convection coefficients. Furthermore, if the number of oversampling layer is chosen properly, we can prove that the convergence rate is proportional to the coarse mesh size. Our analysis also indicates that the size of the oversampling domain weakly depends on the contrast of the heterogeneous coefficients. Several numerical experiments are presented illustrating the performances of our method.