Team Magique-3D

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Section: New Results

Mixed Hybrid Methods for the Helmholtz Equation

Participants : Mohamed Amara, Angela Bernardini ,, Rabia Djellouli.

The standard Finite Element Method (FEM) is based on continuous piecewise polynomials Galerkin approximation. This approach provides a quasi-optimal numerical method for elliptic boundary value problems in the sense that the accuracy of the numerical solution differs only by a constant C from the best approximation obtained from a finite element method. Then this property guarantees good performances of computations at any mesh resolution for the Laplace operator but it can not be preserved for other cases. For example, for the Helmholtz equation, the constant C increases with the wave number as a consequence of a lost of ellipticity as the wave number becomes large. This phenomenon is well-known as a pollution effect [22] . It is due to numerical dispersion errors and FEM are able to cope with large wave numbers only if the mesh resolution is also increased suitably. In order to avoid the pollution effect, numerous discretization techniques have been developed. They include the weak element method for the Helmholtz equation, the Galerkin/least-squares method, the quasi-stabilized finite element method, the partition of unity method, the residual-free bubbles for the Helmholtz equation, the ultra-weak variational method, the least squares method, and recently a discontinuous Galerkin method has been introduced by Farhat et al., [27] , [28] . In this work, we present a new mixed hybrid method in the spirit of [24] for the solution of the Helmholtz equation in the high-frequency regime. Our approach is based on the local approximation of the solution by oscillating finite element polynomials using quadrilateral shaped elements. The shape functions have been chosen in such a way that they oscillate more and more as the wave number increases. In this way the oscillations of the Helmholtz solution are already included in the local approximation of the solution. But the continuity across each interior element interface is lost and it is weakly enforced by introducing suitable Lagrange multipliers which could penalize the computational burden of the method. In fact the discontinuous nature of the approximation enables us to use a static condensation of primal variables prior to assembly. Consequentely, the cost of the new method is determined by the total number of Lagrange multipliers introduced at each edge of the finite element mesh, and by the sparsity pattern of the corresponding system matrix.


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