Explicit and Energy-Conserving Constraint Energy Minimizing Generalized Multiscale Discontinuous Galerkin Method for Wave Propagation in Heterogeneous Media

Abstract

In this work, we propose a local multiscale model reduction approach for the timedomain scalar wave equation in a heterogenous media. A fine mesh is used to capture the heterogeneities of the coefficient field, and the equation is solved globally on a coarse mesh in the discontinuous Galerkin discretization setting. The main idea of the model reduction approach is to extract dominant modes in local spectral problems for representation of important features, construct multiscale basis functions in coarse oversampled regions by constraint energy minimization problems, and perform a Petrov-Galerkin projection and a symmetrization onto the coarse grid. The method is expicit and energy conserving, and exhibits both coarse-mesh and spectral convergence, provided that the oversampling size is appropriately chosen. We study the stability and convergence of our method. We also present numerical results on the Marmousi model in order to test the performance of the method and verify the theoretical results.

Contrast-independent partially explicit time discretizations for multiscale wave problems

“…We have tried some mass lumping schemes; however, their performance were not optimal. We believe for mass lumping other discretizations (mixed or Discontinuous Galerkin, e.g., [5]) may be more effective. Here, we consider one example with mass lumping.…”

Online adaptive algorithm for Constraint Energy Minimizing Generalized Multiscale Discontinuous Galerkin Method

“…Our work is built within the framework of a class of recently developed mutliscale methods, namely the constraint energy minimizing generalized multiscale finite element method (CEM-GMsFEM), which exhibits both coarse mesh convergence and spectral convergence. More precisely, CEM-GMsFEM is considered within a DG discretization setting [9] and extended to wave propagation in heterogeneous and high contrast media [6]. Local spectral problems and constraint energy minimization problems are used to construct multiple multiscale DG basis functions per coarse region, which are then coupled to formulate a global coarse-scale system of equations using the interior penalty discontinuous Galerkin (IPDG) formulation.…”