On different finite element methods for approximating the gradient of the solution to the helmholtz equation

Abstract

We consider the numerical solution of the Helmholtz equation by different finite element methods. In particular, we are interested in finding an efficient method for approximating the gradient of the solution. We first approximate the gradient by the standard Ritz-Galerkin method. As a second method a two-stage method due to Aziz and Werschulz (1980) is presented. It is shown that this method gives the same accuracy in the computed gradient and in the computed solution also in the nonconforming case. Finally, a direct method with asymptotic error estimates is given. It turns out that the presented direct method is of lowest computational complexity. Test examples are presented to illustrate the accuracy of the methods.