The generalized finite element method for Helmholtz equation: Theory, computation, and open problems

In this paper we address the generalized finite element method for the Helmholtz equation. We obtain our method by employing the finite element method on Cartesian meshes, which may overlap the boundaries of the problem domain, and by enriching the approximation by plane waves pasted into the finite element basis at each mesh vertex by the partition of unity method. Here we address the q-convergence of the method, where q is the number of plane waves added at each vertex, for the class of smooth (analytic) solutions for which we get better than exponential convergence for sufficiently small h depending on p. An important observation is that we can monitor the accuracy in any computed solution quantity of interest at negligible cost by using q-extrapolation. Our results assume exact integration of all the employed integrals. Further studies are needed to analyze the effects of the numerical integrations, and also the effect of the roundoff errors.