Mass lumping techniques in the spectral element method: On the equivalence of the row-sum, nodal quadrature, and diagonal scaling methods

Abstract

In the context of wave propagation analysis, the spectral element method (SEM) in conjunction with a diagonal mass matrix is often the method of choice. Therefore, it is of high importance to investigate the influence of different mass lumping schemes on the accuracy of the numerical results. To this end, we compare the performance of three established methods including the row-sum method, the nodal quadrature method, and the diagonal scaling method. The theoretical analysis of these methods reveals a close connection between them. Under certain conditions, that are discussed in detail in this article, we are able to show a direct equivalence between these three approaches. In this regard, the attainable accuracy of the numerical integration of the mass matrix plays an important role. By means of several dynamic benchmark problems we verify the theoretical results and illustrate the convergence properties of the lumped mass matrix SEM in comparison to a formulation based on the consistent mass matrix.