## Section: New Results

Keywords : Discontinous Galerkin Method, Helmholtz Equation, Plane Waves.

### Discontinuous Galerkin methods with plane waves basis for Helmholtz's equation in 3D

Participants : Mohamed Amara, Rabia Djellouli, Magdalena Grigoroscuta.

We have designed a new mixed-hybrid-type solution methodology to be applied for solving high-frequency Helmholtz problems. The proposed approach distinguishes itself from similar methods by a local approximation of the solution with oscillated finite elements polynomials satisfying the wave equation. A weak continuity of the solution across the element interfaces is enforced using Lagrange multipliers. Note that the discontinuous nature of the approximation at the element-level allows to apply static condensation of primal variable prior to assembly. Therefore, the computational cost of the proposed method is reduced, and is mainly dependent on the total number of Lagrange multipliers degrees of freedom, and by the sparsity pattern of the resulting matrix. Hence, the proposed approach combines the features of standard Galerkin finite elements techniques in terms of implementation complexities, and the oscillating aspect of the shape functions needed for approximating waves in the high frequency regime. Preliminary numerical results obtained in the case of two-dimensional wave guides and scattering problems using lower order element (OP41, OP82) clearly illustrate the computational efficiency of the proposed solution methodology. We have also analyzed mathematically the convergence of the proposed method in the case of OP41 element and have established a priori and a posteriori error estimates. We propose here a new discontinuous Galerkin formulation, based on a local approximation of the solution by plane waves that satisfy the wave equation. In order to enforce a weak continuity across the element interfaces, we introduce Lagrange multipliers. The method is built in a variational formulation framework that leads to a linear system associated with a positive definite Hermitian matrix. This matrix results from using a stabilized-like technique. Therefore, we use a preconditioned conjugate gradient algorithm to solve the system without computing the resulting matrix. We have recently applied a regularized-type procedure to the proposed method to address the loss of the stability due to the violation of the inf/sup condition in the case of higher order elements. Consequently the modified mixed hybrid formulation leads to a linear system associated with a positive definite Hermitian matrix.