Effect of Element Distortion on the Numerical Dispersion of Spectral Element Methods

Abstract

Spectral element methods are well established in the field of wave propagation, in particular because they inherit the flexibility of finite element methods and have low numerical dispersion error. The latter is experimentally acknowledged, but has been theoretically shown only in limited cases, such as Cartesian meshes. It is well known that a finite element mesh can contain distorted elements that generate numerical errors for very large distortions. In the present work, we study the effect of element distortion on the numerical dispersion error and determine the distortion range in which an accurate solution is obtained for a given error tolerance. We also discuss a double-grid calculation of the spectral element matrices that preserves accuracy in deformed geometries.

Stability of an explicit high‐order spectral element method for acoustics in heterogeneous media based on local element stability criteria

“…The results in Table are derived for homogeneous material parameters and regular meshes. Numerical tests showing the negative influence of the deformation of quadrilateral elements on the stability are reported in the works of Cohen and Seriani and Oliveira . The influence of distortion is analyzed rigorously in the work of Zhu and Du for simplicial elements and in the work of Askes et al for linear triangles and bilinear quadrilaterals.…”

“…Numerical tests showing the negative influence of the deformation of quadrilateral elements on the stability are reported in the works of Cohen 5 and Seriani and Oliveira. 10,11 The influence of distortion is analyzed rigorously in the work of Zhu and Du 12 for simplicial elements and in the work of Askes et al 13 for linear triangles and bilinear quadrilaterals. Some hints can also be found about the error induced by the presence of a discontinuity or heterogeneity of the material properties, 5,14-16 but general stability criteria have only been derived for specific finite-difference schemes.…”

Spectral Element Approximation of Fredholm Integral Eigenvalue Problems

“…The tensor product of the 1D spectral elements above naturally lead to 2D and 3D basis functions on square/cubic meshes (see [6] for more general geometries).…”

“…This class of methods has been successful on wave propagation problems mainly because they are flexible to deal with complex geometries and produce low dispersion error [6]. This work considers the use of these methods in the solution of the Fredholm integral equation D C(x, y)φ k (y) dy = λ k φ k (x), k = 1, 2, .…”

Some illustrative examples of the use of a spectral-element method in ocean acoustics

“…Very distorted mesh elements can be accurately handled. 5 Complex models that include fluid, elastic, viscoelastic, anisotropic, or porous media 6 can be modeled, making the SEM a method of choice for the numerical modeling of wave propagation through various types of media encountered in ocean acoustics. Furthermore, the calculation of sensitivity kernels can be performed based on adjoint modeling.…”