Introduction to the spectral element method for three-dimensional seismic wave propagation

Dimitri Komatitsch, Jeroen Tromp

Research output: Contribution to journalArticlepeer-review

1182 Scopus citations


We present an introduction to the spectral element method, which provides an innovative numerical approach to the calculation of synthetic seismograms in 3-D earth models. The method combines the flexibility of a finite element method with the accuracy of a spectral method. 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. We illustrate the great potential of the method by comparing it to a discrete wavenumber/reflectivity method for layer-cake models. Both body and surface waves are accurately represented, and the method can handle point force as well as moment tensor sources. For a model with very steep surface topography we successfully benchmark the method against an approximate boundary technique. For a homogeneous medium with strong attenuation we obtain excellent agreement with the analytical solution for a point force.

Original languageEnglish (US)
Pages (from-to)806-822
Number of pages17
JournalGeophysical Journal International
Issue number3
StatePublished - Dec 1999

All Science Journal Classification (ASJC) codes

  • Geophysics
  • Geochemistry and Petrology


  • Attenuation
  • Finite element methods
  • Numerical techniques
  • Seismic modelling
  • Seismic wave propagation
  • Topography


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