Abstract
We analyse the dispersion properties of two types of explicit finite element methods for modelling acoustic and elastic wave propagation on tetrahedral meshes, namely mass-lumped finite element methods and symmetric interior penalty discontinuous Galerkin methods, both combined with a suitable Lax–Wendroff time integration scheme. The dispersion properties are obtained semi-analytically using standard Fourier analysis. Based on the dispersion analysis, we give an indication of which method is the most efficient for a given accuracy, how many elements per wavelength are required for a given accuracy, and how sensitive the accuracy of the method is to poorly shaped elements.
Highlights
Realistic wave propagation problems often involve large three-dimensional domains consisting of heterogeneous materials with complex geometries and sharp interfaces
J Sci Comput (2018) 77:372–396 problems requires a numerical method that is efficient in terms of computation time and is flexible enough to capture the effect of a complex geometry
Obtaining a high quality mesh is quintessential. While both hexahedral and tetrahedral elements are commonly used for threedimensional problems, tetrahedral elements have a big advantage in this respect, since they offer more geometric flexibility and since robust tetrahedral mesh generators based on the Delaunay criterion are available [29,35]
Summary
Realistic wave propagation problems often involve large three-dimensional domains consisting of heterogeneous materials with complex geometries and sharp interfaces. The advantage of finite element methods based on the second-order formulation is that they do not need to compute or store the auxiliary variables that appear in the first-order formulation They can be combined with a leap-frog or higher-order Lax–Wendroff time integration scheme that only requires K stages for a 2K -order accuracy. The dispersion properties of DG methods based on the first-order formulation of the wave problem have already been analysed for Cartesian meshes [1,18], triangles [18,22], and tetrahedra [19]. A dispersion analysis of the mass-lumped finite element and SIPDG methods for tetrahedra is, to the best of our knowledge, still missing, even though most realistic wave problems involve three-dimensional domains for which tetrahedral elements are suitable.
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