Abstract
We present a comprehensive review of Discontinuous Galerkin Spectral Element (DGSE) methods on hybrid hexahedral/tetrahedral grids for the numerical modeling of the ground motion induced by large earthquakes. DGSE methods combine the exibility of discontinuous Galerkin meth-ods to patch together, through a domain decomposition paradigm, Spectral Element blocks where high-order polynomials are used for the space discretization. This approach allows local adaptivity on discretization parameters, thus improving the quality of the solution without affecting the compu-tational costs. The theoretical properties of the semidiscrete formulation are also revised, including well-posedness, stability and error estimates. A discussion on the dissipation, dispersion and stability properties of the fully-discrete (in space and time) formulation is also presented. Here space dis-cretization is obtained based on employing the leap-frog time marching scheme. The capabilities of the present approach are demonstrated through a set of computations of realistic earthquake scenar-ios obtained using the code SPEED (http://speed.mox.polimi.it), an open-source code specifically designed for the numerical modeling of large-scale seismic events jointly developed at Politecnico di Milano by The Laboratory for Modeling and Scientific Computing MOX and by the Department of Civil and Environmental Engineering.
Highlights
In the last decades, the scientific research on elastic waves propagation problems modeled through theelastodynamics equation has experienced a constantly increasing interest in the mathematical, geophysical and engineering communities
We consider a bounded domain Ω ⊂ R3, and assume that its boundary is decomposed into three disjoint portions ΓD, ΓN and ΓNR, where we impose the values of displacement, of tractions, and of the fictitious tractions introduced to avoid unphysical reflections, respectively
We refer the reader to the relevant publication and to the SCEC Broadband Platform for the documentation of the code
Summary
The scientific research on elastic waves propagation problems modeled through the (visco)elastodynamics equation has experienced a constantly increasing interest in the mathematical, geophysical and engineering communities. A similar approach has been presented for spectral elements and DGSE approximations on triangular grids [76,78], where different sets of interpolating nodes have been compared, and in [15], where the authors have used the modal boundary adapted functions proposed in [106] All these works deal with two-dimensional model problems and show that triangular spectral elements feature dispersion and dissipation properties similar to those of the standard tensor product spectral elements. The aim of this work is to present a comprehensive review on numerical modeling of seismic waves based on DGSE methods on hybrid hexahedral/tetrahedral grids These methods combine the flexibility of discontinuous Galerkin methods to connect together, through a domain decomposition paradigm, Spectral Element blocks where high-order polynomials are used. Some technical results needed for the theoretical analysis are contained in Appendix A
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