A discontinuous Galerkin based multiscale method for heterogeneous elastic wave equations

TL;DR

This paper introduces a multiscale discontinuous Galerkin method for efficiently simulating elastic wave equations in highly heterogeneous media, combining spectral analysis and energy minimization for accurate coarse-scale modeling.

Contribution

It develops a novel multiscale basis construction using spectral problems and energy minimization within a discontinuous Galerkin framework for elastic wave equations.

Findings

Method achieves coarse-mesh and spectral convergence.

Numerical results confirm stability and accuracy.

Explicit energy-conserving scheme for fast simulations.

Abstract

In this paper, we develop a local multiscale model reduction strategy for the elastic wave equation in strongly heterogeneous media, which is achieved by solving the problem in a coarse mesh with multiscale basis functions. We use the interior penalty discontinuous Galerkin (IPDG) to couple the multiscale basis functions that contain important heterogeneous media information. The construction of efficient multiscale basis functions starts with extracting dominant modes of carefully defined spectral problems to represent important media feature, which is followed by solving a constraint energy minimization problems. Then a Petrov-Galerkin projection and systematization onto the coarse grid is applied. As a result, an explicit and energy conserving scheme is obtained for fast online simulation. The method exhibits both coarse-mesh and spectral convergence as long as one appropriately chose the oversampling size. We rigorously analyze the stability and convergence of the proposed method. Numerical results are provided to show the performance of the multiscale method and confirm the theoretical results.