Constraint Energy Minimizing Generalized Multiscale Finite Element Method for Convection Diffusion Equations with Inhomogeneous Boundary Conditions

TL;DR

This paper introduces a new multiscale finite element method for convection-diffusion equations with complex boundary conditions and high-contrast coefficients, achieving accurate error convergence and handling nonlinear problems.

Contribution

The paper develops the CEM-GMsFEM for convection-diffusion equations with inhomogeneous boundary conditions, including boundary correctors and schemes for time-dependent problems, with proven error bounds.

Findings

First-order convergence in energy norm

Second-order convergence in L2 norm

Effective handling of nonlinear problems with Strang splitting

Abstract

In this paper, we develop the constraint energy minimizing generalized multiscale finite element method (CEM-GMsFEM) for convection-diffusion equations with inhomogeneous Dirichlet, Neumann and Robin boundary conditions, along with high-contrast coefficients. For time independent problems, boundary correctors $\mathcal{D}^m$ and $\mathcal{N}^{m}$ for Dirichlet, Neumann, and Robin conditions are designed. For time dependent problems, a scheme to update the boundary correctors is formulated. Error analysis in both cases is given to show the first-order convergence in energy norm with respect to the coarse mesh size $H$ and second-order convergence in $L^2-$norm, as verified by numerical examples, with which different finite difference schemes are compared for temporal discretization. Nonlinear problems are also demonstrated in combination with Strang splitting.