A Priori Energy Estimates for a Class of Dg Approximations in Elastodynamics

Abstract eng:
Discontinuous Galerkin (DG) methods were introduced in the 70’s for approximating pure hyperbolic problems. Over the last decade, DG methods have been improved and generalizad to virtually any kind of partial differential equations, mainly because of their stability properties and their flexibility to account for both grid and polynomial refinements. In this work we introduce a family of semidiscrete-DG methods for the approximation of a general elastodynamics problem, recovering some of the DG schemes frequently used in elastodynamics like the Interior Penalty (IP) method or the Local Discontinuous Galerkin (LDG) method. Using the energy of the system, we prove stability for the resulting DG schemes and perform the apriori error analysis of the methods. We also propose some schemes that preserve the total energy of the system. After validating these results on some test cases, we present computations obtained on realistic seismic wave propagation problems.

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