Exact integration scheme for planewave-enriched partition of unity finite element method to solve the Helmholtz problem

In this paper, we present an exact integration scheme to compute highly oscillatory integrals that appear in the solution of the two-dimensional Helmholtz problem using the planewave-enriched partition of unity finite element method. In the proposed scheme, such oscillatory integrals are computed by a recursive application of the divergence theorem, eventually expressing the integrals in terms of evaluations of the corresponding integrands at the nodes of the finite element mesh. The number of such function evaluations is independent of the wave number k, which permits the scheme to be used for arbitrary high values of k. We consider finite element meshes with unstructured triangular and structured rectangular elements, and present numerical results for three canonical benchmark Helmholtz problems to demonstrate the accuracy and efficacy of the method.