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In this paper we study the -version of the Partition of Unity Method for the Helmholtz equation. The method is obtained by employing the standard bilinear finite element basis on a mesh of quadrilaterals discretizing the domain as the Partition of Unity used to paste together local bases of special wave-functions employed at the mesh vertices. The main topic of the paper is the comparison of the performance of the method for two choices of local basis functions, namely a) plane-waves, and b) wave-bands....
We are concerned with a finite element approximation for time-harmonic wave
propagation governed by the Helmholtz equation. The usually oscillatory behavior of
solutions, along with numerical dispersion, render standard finite element methods
grossly inefficient already in medium-frequency regimes. As an alternative, methods
that incorporate information about the solution in the form of plane waves have
been proposed. We focus on a class of Trefftz-type discontinuous Galerkin methods that
...
In this paper we propose a method for improving the convergence rate of the mixed finite element approximations for the Stokes eigenvalue problem. It is based on a postprocessing strategy that consists of solving an additional Stokes source problem on an augmented mixed finite element space which can be constructed either by refining the mesh or by using the same mesh but increasing the order of the mixed finite element space.
We present a weak parametrix of the operator of the CFIE equation. An interesting feature of this parametrix is that it is compatible with different discretization strategies and hence allows for the construction of efficient preconditioners dedicated to the CFIE. Furthermore, one shows that the underlying operator of the CFIE verifies an uniform discrete Inf-Sup condition which allows to predict an original convergence result of the numerical solution of the CFIE to the exact one.
On exhibe dans cette note une paramétrix (au sens faible) de l'opérateur
sous-jacent à l'équation CFIE de l'électromagnétisme. L'intérêt de cette
paramétrix est de se prêter à différentes stratégies de discrétisation
et ainsi de pouvoir être utilisée comme préconditionneur de la CFIE.
On montre aussi que l'opérateur sous-jacent à la CFIE satisfait une condition
Inf-Sup discrète uniforme, applicable aux espaces de discrétisation usuellement rencontrés
en électromagnétisme, et qui permet d'établir...
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