Discontinuous Galerkin Finite Element Method for the Wave Equation

By Marcus J. Grote, Anna Schneebeli, Dominik Schötzau. SIAM Journal on Numerical Analysis, 2006

346 citations recorded by the source corpus.

The symmetric interior penalty discontinuous Galerkin finite element method is presented for the numerical discretization of the second‐order wave equation. The resulting stiffness matrix is symmetric positive definite, and the mass matrix is essentially diagonal; hence, the method is inherently parallel and leads to fully explicit time integration when coupled with an explicit time‐ stepping scheme. Optimal a priori error bounds are derived in the energy norm and the $L^2$‐norm for the semidiscrete formulation. In particular, the error in the energy norm is shown to converge with the optimal order ${\cal O}(h^{\min\{s,\ell\}})$ with respect to the mesh size h, the polynomial degree $\ell$, and the regularity exponent s of the continuous solution. Under additional regularity assumptions, the $L^2$‐error is shown to converge with the optimal order ${\cal O}(h^{\min\{s,\ell\}})$. Numerical results confirm the expected convergence rates and illustrate the versatility of the method.

Subfields: Computational Mechanics, Electrical and Electronic Engineering, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/a6c915f673101e6fd4781d805f091d543a41db93

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Mechanics

In Electrical and Electronic Engineering

In Numerical Analysis

Citations in the corpus

Energy conserving local discontinuous Galerkin methods for wave propagation problems (2013, methodology): “Usually it is difficult to obtain DG schemes for wave equations which are non-dissipative and optimal high order accurate, without going to staggered mesh [5, 6] or using penalty with a parameter to adjust [18, 19].”
Summation by Parts Operators for Finite Difference Approximations of Second-Derivatives with Variable Coefficients (2012, background): “In many applications, such as general relativity [28, 3], seismology [12, 30], oceanography [21], acoustics [29, 22, 5, 2, 6] and electromagnetics [31, 7], the underlying equations are systems of second-order hyperbolic partial differential equations.”
High-order finite-element seismic wave propagation modeling with MPI on a large GPU cluster (2010, methodology): “As mentioned in the Introduction, discontinuous Galerkin methods have also been applied to the modeling of seismic waves on CPUs [31–37]; therefore it is likely that based on the ideas of Klöckner et al. [47] to use such techniques on GPUs for Maxwell’s equations they could be adapted to the modelin”
Forward and adjoint simulations of seismic wave propagation on fully unstructured hexahedral meshes (2011, background): “…Hill 1973; Arnold 1982; Falk & Richter 1999; Hu et al. 1999; Cockburn et al. 2000; Giraldo et al. 2002; Riviére & Wheeler 2003; Monk & Richter 2005; Grote et al. 2006; Ainsworth et al. 2006; Bernacki et al. 2006; Dumbser & Käser 2006; De Basabe et al. 2008; de la Puente et al. 2009; Wilcox et…”