| The objectives of this research are to develop adaptive finite element methods for the Helmholtz equation in exterior domains, and to implement these methods for problems of time-harmonic exterior acoustics. The numerical solution of this equation is made difficult, however, by the presence of a term involving the wave number, which leads to indefiniteness of the operator and stringent mesh size requirements for propagating solutions. The aim is to demonstrate that adaptivity can be a powerful tool for achieving significant reductions in the cost of solving this equation using the finite element method.; Finite element computations in exterior domains are enabled by applying the Dirichlet-to-Neumann (DtN) condition on an artificial exterior boundary. Two finite element formulations are considered: The Galerkin and Galerkin Least-Squares (GLS) methods. The present work focuses on development of an a posteriori error estimator for computing the solution error distribution, and an h-adaptive strategy for specifying mesh size refinement. It is shown that these methods lead to an efficient adaptive mesh. Solutions are computed for two model problems of acoustics: Plane wave scattering by a rigid infinite cylinder, and nonuniform radiation from an infinite cylinder. Detailed studies with respect to CPU and computer storage costs are performed in the context of a direct solver for the linear system of equations. For the nonuniform radiation problem, the adaptive mesh is twenty times more CPU-cost effective than a uniform mesh (for the Galerkin formulation). When coupled with the GLS formulation, the adaptive mesh is forty times more efficient than Galerkin computations on a uniform mesh.; Finally, a methodology for computing error estimates (not just relative error indicators) is established through calculation of the scaling constant appearing in the error estimator. Several measures are computed to assess the quality of the error estimator. Results indicate that the error distributions are adequately captured. However, the quality of the global error estimates degrades with increasing wave number. |
