A locally discontinuous enriched finite element formulation for acoustics

Abstract

SUMMARYIn (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 0 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.

Numerical Analysis of a Locally Projected Discontinuous Galerkin Method for Elliptic Problems

“…A great variety of Discontinuous Galerkin (DG) methods have been proposed and analyzed over the last decades for elliptic [7,15,3,4,14,16,27,8,26,5], parabolic [2,25] and hyperbolic [24,22,18,19,1,20,21] problems. Robustness, flexibility for implementing h and p-adaptivity strategies and easy parallelization are well known advantages of DG methods arising from the use of broken finite element spaces.…”

Discontinuous subgrid formulations for transport problems

“…Thus, some additional stabilization may be necessary, depending on the problem. Some approaches to address this issue are based on slope limiters [12][13][14], Petrov-Galerkin stabilizations [15,16], bubble stabilization [17,18], interior penalty-type stabilizations [16,19] and subgrid stabilization [1]. The subgrid method introduced in the latter work is a combination of a DG method with a linear eddy viscosity model, which is also controlled by a user-defined mesh dependent coefficient m T .…”

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

“…Numerical approximation of time-harmonic acoustic, elastic and electromagnetic wave problems governed by the Helmholtz equation is particularly challenging as reported in a vast literature [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. The oscillatory behavior of the exact solution and the quality of the numerical approximation depend on the wave number k. To approximate Helmholtz equation with acceptable accuracy the resolution of the mesh should be adjusted to the wave number according to a rule of thumb [1], which prescribes a minimum number of elements per wavelength.…”

“…Finite element methods based on variational 0045-7825/$ -see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.cma.2007.11.001 formulations, such as Residual-Based Finite Element Method (RBFEM) [7] and Discontinuous Finite Element Method at Element Level (DGB) [8,9], have also been developed to minimize the phase error in two-dimensions.…”