Galerkin/least-squares finite element methods for the reduced wave equation with non-reflecting boundary conditions in unbounded domains

Abstract

Finite element methods are constructed for the reduced wave equation in unbounded domains. Exterior boundary conditions for a computational problem are derived from an exact relation between the solution and its derivatives on an artificial boundary by the DtN method, precluding singular behavior in finite element models. Galerkin and Galerkin/least-squares finite element methods are presented. Model problems of radiation with inhomogeneous Neumann boundary conditions in plane and spherical configurations are employed to design and evaluate the numerical methods in the entire range of propagation and decay. The Galerkin/least-squares method with DtN boundary conditions is designed to exhibit superior behavior for problems of acoustics, providing accurate solutions with relatively low mesh resolution and allowing numerical damping of unresolved waves. General convergence results guarantee the good performance of Galerkin/least-squares methods on all configurations of practical interest. Numerical tests validate these conclusions.