Spectral-element method with an optimal mass matrix for seismic wave modelling

The spectral-element method (SEM), which combines the flexibility of the finite element method (FEM) with the accuracy of spectral method, has been successfully applied to simulate seismic wavefields in geological models on different scales. One kind of SEMs that adopts orthogonal Legendre polynomials is widely used in seismology community. In the SEM with orthogonal Legendre polynomials, the Gauss-Lobatto-Legendre (GLL) quadrature rule is employed to calculate the integrals involved in the SEM leading to a diagonal mass matrix. However, the GLL quadrature rule can exactly approximate only integrals with a polynomial degree below 2N-1 (N is the interpolation order in space) and cannot exactly calculate those of polynomials with degree 2N involved in the mass matrix. Therefore, the error of the mass matrix originating from inexact numerical integration may reduce the accuracy of the SEM. To improve the SEM accuracy, we construct a least-squares objective function in terms of numerical and exact integrals to increase the accuracy of the GLL quadrature rule. Then, we utilise the conjugate gradient method to solve the objective function and obtain a set of optimal quadrature weights. The optimal mass matrix can be obtained simultaneously by utilising the GLL quadrature rule with optimal integration weights. The improvement in the numerical accuracy of the SEM with an optimal mass matrix (OSEM) is demonstrated by theoretical analysis and numerical examples.