Numerical investigation of the edge-based gradient smoothing technique for exterior Helmholtz equation in two dimensions

In order to improve the accuracy of the standard finite element method (FEM) solutions for Helmholtz equation and reduce the numerical error, an edge-based gradient smoothing technique (E-GST) is incorporated into the standard finite element method to give an ES-FEM for the exterior Helmholtz equation in two dimensions. In the ES-FEM, the smoothing domains are formed by sequentially linking the terminal points of each edge and the centers of the neighboring elements. Then the E-GST is used to smooth the gradient on these smoothing domains. Due to the softening effect of the E-GST, the “overly-stiff” property of the original finite element model can be properly softened and the accuracy of the solution will be improved significantly. With the aim to handle the exterior Helmholtz problems in infinite domain, we introduce an artificial boundary B to obtain a finite computational domain and the Dirichlet-to-Neumann (DtN) map is used to guarantee that all the waves on B are outgoing. Several typical numerical examples are considered to assess the effectivity and feasibility of the ES-FEM with the DtN map. It is shown that the present model works well for exterior Helmholtz equation and can produce better solutions than standard FEM.