The Spectral Element Method (SEM): Formulation and Implementation of the Method for Engineering Seismology Problems

Abstract eng:
Various numerical methods have been proposed and used to investigate wave propagation in realistic earth media. Recently an innovative numerical method, known as the Spectral Element Method (SEM), has been developed and used in connection with wave propagation problems in 3D elastic media (Komatitsch and Tromp, 1999). The SEM combines the flexibility of a finite element method with the accuracy of a spectral method and it is not cumbersome in dealing with non-flat free surface and spatially variable anelastic attenuation. The SEM is a highly accurate numerical method that has its origins in computational fluid dynamics. One uses a weak formulation of the equations of motion, which are solved on a mesh of hexahedral elements that is adapted to the free surface and to the main internal discontinuities of the model. The wavefield on the elements is discretized using high-degree Lagrange interpolants, and integration over an element is accomplished based upon the Gauss-Lobatto-Legendre integration rule. This combination of discretization and integration results in a diagonal mass matrix, which greatly simplifies the algorithm. The most important property of the SEM is that the mass matrix is exactly diagonal by construction, which drastically simplifies the implementation and reduces the computational cost because one can use an explicit time integration scheme without having to invert a linear system. Furthermore, it allows an efficient parallel implementation. We sketch the formulation of the SEM in matrix form that is familiar to earthquake engineers. We demonstrate the efficiency and effectiveness of the method by implementing the proposed formulation to study various simple/canonical problems.

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