Computational methods for wave propagation problems in unbounded domains

In this dissertation I have proposed a novel fictitious domain method based on a distributed Lagrange multiplier for the solution of the time-dependent problem of scattering by an obstacle. I have introduced the fictitious domain method for the case of the two-dimensional scalar wave equation, as well as for the two-dimensional transverse magnetic (TM) mode of Maxwell's equations, along with a Dirichlet condition on the boundary of the obstacle in each case.; In the case of the two-dimensional scalar wave equation, I have presented a symmetrized operator splitting scheme to decouple the operator that propagates the wave and the operator that enforces the Dirichlet condition on the boundary of the obstacle. I have studied different discretizations for the different subproblems involved in the operator splitting scheme. These include conforming finite elements as well as mixed finite element formulations utilizing the lowest order Nédélec edge elements on rectangular grids. I have presented an analysis of the fictitious domain approach and the symmetrized operator splitting scheme for a one-dimensional wave problem. Comparisons are performed with other relevant numerical schemes, such as the finite difference scheme, that show the advantages of the formulation proposed in this thesis.; I have constructed a mixed finite element formulation for the two-dimensional TM mode of the uniaxial perfectly matched layer (PML) for Maxwell's equations. Energy estimates that demonstrate the well-posedness of the model are presented. I have employed a mixed discretization which utilizes the lowest order Raviart-Thomas elements and bilinear nodal finite elements on rectangular grids. I have performed a plane wave analysis to study the errors that arise due to dispersion, anisotropy, the numerical discretization as well as the termination of the PML by a perfect conductor condition. Finally, I have incorporated the fictitious domain approach into the mixed finite element model for the PML.; Numerical results that validate the effectiveness of the different models are presented in this dissertation.