A nearly optimal Galerkin projected residual finite element method for Helmholtz problem

A Finite Element Formulation for scalar and linear second-order boundary value problems is introduced. The new method relies on a variational formulation obtained following the usual path of appending to the Galerkin variational formulation, a balanced residual form of the governing partial differential equation computed within each element. The novelty consists of projecting the residual in a subspace defined for each element, which gives rise to the name of the method: Galerkin Projected Residual (GPR). This subspace is built by systematically exploring some a priori criteria (either based on the physics or on the underlying mathematics). The method can be used to stabilize a variety of problems. Here it is applied to Helmholtz equation, where standard Galerkin formulations are known to present poor approximations for high wave numbers. The method is formally introduced along with some numerical examples that are used to assess the improvements achieved.