h-adaptive finite element computation of time-harmonic exterior acoustics problems in two dimensions

In this paper we carry out adaptive finite element computations of the Helmholtz equation in two dimensions, in the context of time-harmonic exterior acoustics. The purpose is to demonstrate potentially significant cost savings engendered through adaptivity for propagating solutions at moderate wave numbers (ka = 2 pi, 4 pi). The computations are performed on meshes of linear triangles, and are adapted to the solution by locally changing element sizes (h-refinement). The adaptive procedure involves applying an explicit, residual-based a posteriori error estimator and an h-adaptive strategy, which were derived in a previous paper. The adaptive meshes are then obtained through global mesh regeneration using an advancing front mesh generator. Two different finite element formulations are considered: The Galerkin and Galerkin Least-Squares (GLS) methods. The infinite physical domain is truncated by an artificial exterior boundary, on which a fully coupled Dirichlet-to-Neumann (DtN) boundary condition is applied. Two different problems are computed: Plane wave scattering by a rigid infinite surface (circular and square cross sections), and nonuniform radiation from an infinite circular cylinder. Detailed cost studies with respect to an active column direct solver are performed. For the nonuniform radiation problem, the adaptive mesh is twenty times more cost effective than a uniform mesh (for Galerkin). When coupled with the GLS formulation, the adaptive mesh is forty times more efficient than Galerkin computations on a uniform mesh.