Dispersion Analysis of Finite-element Schemes for a First-order Formulation of the Wave Equation

Abstract

SUMMARYWe investigated one-dimensional numerical dispersion curves and error behaviour of four finite-element schemes with polynomial basis functions: the standard elements with equidistant nodes, the LegendreGauss-Lobatto points, the Chebyshev-Gauss-Lobatto nodes without a weighting function and with. Mass lumping, required for efficiency reasons and enabling explicit time stepping, may adversely affect the numerical error. We show that in some cases, the accuracy can be improved by applying one iteration on the full mass matrix, preconditioned by its lumped version. For polynomials of degree one, this improves the accuracy from second to fourth order in the element size. In other cases, the improvement in accuracy is less dramatic.

Should We Use the First- or Second-order Formulation with Spectral Elements for Seismic Modelling?

“…Since inversion of this large sparse matrix is costly, mass lumping can be applied, but then the superior dispersion behaviour is lost. For that reason, Shamasundar et al (2015) proposed the use of defect correction as an efficient alternative. In 1D, we found that one iteration with the consistent mass matrix, preconditioned by the mass-lumped mass matrix, reduced the numerical dispersion error.…”