A locally discontinuous enriched finite element formulation for acoustics

In (Comput. Methods Appl. Mech. Eng. 2006, in press) we introduced a discontinuous Galerkin finite element method for Helmholtz equation in which continuity is relaxed locally in the interior of the element. The shape functions associated with interior nodes of the element are bilinear discontinuous bubbles, and the corresponding degrees of freedom can be eliminated at element level by static condensation yielding a global finite element method with the same connectivity of classical C° Galerkin finite element approximations. Stability is provided by the discontinuous bubbles with appropriate choice of the stabilization parameters related to the weak enforcement of continuity inside each element. In the present work, departing from the stencil obtained by condensation of the bubble degrees of freedom, we build a new strategy for determining the optimal values of these parameters aiming at matching the exact wave number in two different directions. Stability and accuracy of the proposed formulation are demonstrated in several numerical examples.