Numerical solution of the Helmholtz Equation using high-order continuous Galerkin methods

Abstract

We show a continuous Galerkin formulation with high-order polynomials to solve the Helmholtz equation. High-order formulations obtain solutions with less numerical error, and can use curved high-order meshes to approximate the domain. To reduce the computational requirements of the high-order formulation, we apply a static condensation technique. Using this technique, we eliminate a set of unknowns from the global linear system and therefore, we solve a smaller system of equations. Then, we recover the full solution by solving several systems that only involve the unknowns of a single element. In the examples we show that the proposed implementation converges optimally to the analytical solution both for two and three dimensional examples. We also show an application where the mesh is curved in order to better capture the geometry.