| Here, we develop and analyze a new class of spectral element methods for the simulations of elastic wave equation. The equation of poroelastic wave equation in bi-phase medium are also being considered in this thesis. The spatial discretization and the choice of interpolation nodes are the major components of the spectral element methods. The spatial discretization is based on piecewise polynomial approximation defined on staggered grids. The resulting method combines the advantages of both staggered-grid based methods and classical non-staggered-grid based spectral element methods. The method is energy conserving and does not require the use of any numerical flux, because of the staggered local continuity of the basis functions. Our new method also uses Gauss-Radau points as interpolation nodes, and the resulting mass matrix is diagonal, thus time marching is explicit and is very efficient. Moreover, for the elastic wave equation, we give a rigorous proof for the optimal convergence of the method. In terms of dispersion, we present a numerical study for the numerical dispersion and show that this error is of very high order. Next, we develop a mortar formulation to combine the staggered discontinuous method with triangular SDG method and the spectral element methods. In particular, a fine triangular mesh is used near the free surface and a coarse mesh is used in the rest of the domain. The two meshes are in general not match. The continuity of the velocity on the interface of the two domain is enforced by a Lagrange multiplier. The method is very efficient. Meanwhile, some numerical results about convergence tests and applications to unbounded domain problems with perfectly matched layer are shown. We also apply our spectral element method to the poroelastic Biot's equation in bi-phase medium. We provide various numerical results to show that our scheme is also applicable to the poroelastic Biot's equation. |
