A residual-based finite element method for the Helmholtz equation

A new residual-based nite element method for the scalar Helmholtz equation is developed. This method is obtained from the Galerkin approximation by appending terms that are proportional to residuals on element interiors and inter-element boundaries. The inclusion of residuals on inter-element boundaries distinguishes this method from the well-known Galerkin least-squares method and is crucial to the resulting accuracy of this method. In two dimensions and for regular bilinear quadrilateral nite elements, it is shown via a dispersion analysis that this method has minimal phase error. Numerical experiments are conducted to verify this claim as well as test and compare the performance of this method on unstructured meshes with other methods. It is found that even for unstructured meshes this method retains a high level of accuracy.