A comparative study of the boundary and finite element methods for the Helmholtz equation in two dimensions

The performance of the boundary and finite element methods for the Helmholtz equation in two dimensions is investigated. To facilitate the comparison, the system of linear equations arising from the finite element formulation is reduced to a smaller system involving the boundary values of the unknown function and its normal derivative alone. The difference between the boundary and finite element solutions is then expressed in terms of a difference matrix operating on the boundary data. Numerical investigations show that the boundary element method is generally more accurate than the finite element method when the size of the finite elements is comparable to that of the boundary elements, especially for the Dirichlet problem where the boundary values of the solution are specified. Exceptions occur in the neighborhood of isolated points of the Helmholtz constant where eigenfunctions of the boundary integral equation arise and the boundary element method fails to produce a unique solution.