Weak Galerkin Method for the Helmholtz Equation with DtN Boundary Condition

Abstract

In this article, we consider a weak Galerkin finite element method for the two dimensional exterior Helmholtz problem. After introducing a nonlocal boundary condition by means of the exact Dirichlet to Neumann (DtN) operator for the exterior problem, we prove that the existence and uniqueness of the weak Galerkin finite element solution for this problem. Then, applying some projection techniques, we establish a priori error estimate, which include the effect of truncation of the DtN boundary condition as well as the spatial discretization. Finally, some numerical examples are presented to confirm the theoretical predictions.