A Constraint energy minimizing generalized multiscale finite element method for parabolic equations

TL;DR

This paper introduces a new multiscale finite element method for parabolic equations with multiscale coefficients, providing rigorous convergence analysis and demonstrating effectiveness through numerical experiments.

Contribution

The paper develops the CEM-GMsFEM for parabolic equations, with convergence rates independent of media scale and contrast, and includes a posteriori error estimation.

Findings

Convergence rate is first order in energy norm and second order in L2 norm.

Method performance is validated through numerical experiments on porous media.

Convergence is independent of media scale length and contrast.

Abstract

In this paper, we present a Constraint Energy Minimizing Generalized Multiscale Finite Element Method (CEM-GMsFEM) for parabolic equations with multiscale coefficients, arising from applications in porous media. We will present the construction of CEM-GMsFEM and rigorously analyze its convergence for the parabolic equations. The convergence rate is characterized by the coarse grid size and the eigenvalue decay of local spectral problems, but is independent of the scale length and contrast of the media. The analysis shows that the method has a first order convergence rate with respect to coarse grid size in the energy norm and second order convergence rate with respect to coarse grid size in $L^2$ norm under some appropriate assumptions. For the temporal discretization, finite difference techniques are used and the convergence analysis of full discrete scheme is given. Moreover, a posteriori error estimator is derived and analyzed. A few numerical results for porous media applications are presented to confirm the theoretical findings and demonstrate the performance of the approach.