Finite Element Methods for Seismic Modelling

In this dissertation, new and more efficient finite element methods for modelling seismic wave propagation are presented and analysed. Seismic modelling is a useful tool for better understanding seismic behaviour in complex rock structures, but it is also a key aspect of full waveform inversion, which is a powerful technique for imaging the structure of the earth's subsurface. The great advantage of finite element methods over other wave modelling methods, like the popular finite difference method, is that it accurately captures the effect of complex topographies, such as mountainous areas and rough seabeds, without refining the grid resolution. Even so, these methods require a huge amount of computational power and making them more efficient is of great value in many industrial applications. Topics addressed in this dissertation include: new and sharper bounds for the penalty term and time step size of the Discontinuous Galerkin method, new and significantly more efficient mass-lumped tetrahedral elements, new and efficient quadrature rules for evaluating the stiffness matrix of these mass-lumped elements, stability properties of a basic local time-stepping algorithm, and a dispersion analysis and comparison of multiple finite element methods. Overall, the finite element methods and algorithms presented in this dissertation allow for a much faster modelling of seismic waves. This is especially true for the new mass-lumped finite elements, which in some cases result in a speed of a factor 10 compared to other finite element methods. These improvements make the use of finite element methods much more attractive for geophysical applications or other industrial applications that involve solving wave propagation problems