Formulações Galerkin descontínuo-contínuo para o problema de Helmholtz

Abstract

In this work, we propose two consistente Continous-Discontinous Galerkin formulations for the Helmholtz problem: a hybridized Discontinuous Galerkin formulation ( HDG method) that Works with a continuous trace space and a Continuous- Discontinuous Galerkin formulation ( CDG method) that Works with a continuous and a discontinuous component. For the HDG formulation, we present a static condensation analysis where we obtain a global system that smaller than the global system generated by other current hybrid methods. Furthermore, we show that the HDG formulation is a well-posed problem from a certain degree of mesh refinement. For the CDG formulation, we present a formulation where a stability transfer bilinear form is used internally connecting the discontinuous component as a functions of continuous component which which makes the problem a locally well-posed continuous problem. As numerical expriments, we present the computacional time for the methods HDG and CDG compared with the computacional time of a continuous Galerkin approach ( CG) for a fixed polynomial approximation and different mesh refinements. Numerical results show that the CDG method is the most robust and accurate,despist have a computacional effort higher than the others. The HDG method has a computacional time similar to that of the CG method. However, it needs less mesh refinement to converge to the exact solution. Finally, we conclude that each method has distinct advantages, but both have great potential to be explored.