High Order Finite Element Method for solving Convected Helmholtz equation in radial and axisymmetric domains. Application to Helioseismology

Abstract : This document concerns the solution of convected Helmholtz equation for radial or axisymmetric configuration. After setting the considered problem and the associated equations, we propose a numerical method adapted for geometries that only vary radially. Then we propose a numerical method for geometries with an axial symmetry. In these two situations, the solution is computed with finite elements after having been decomposed on a basis of orthogonal modes : spherical harmonics in the radial geometry and Fourier modes in the axisymmetric geometry. The number of required modes depends on the configuration (and especially location) of the source. The two methods are tested on the computation of Green's functions for which analytical solutions are available for quantitative comparison. The methods are then compared with the classical 3D finite elements method, and the performances are assessed for an academic test case, showing the advantages of each method in terms of computation time and memory usage. In the context of helioseismology, the perturbation of the fluid displacement in the sun can be modeled accurately by Galbrun's equations. We show that under some assumptions on the background medium, the Galbrun's equations can be simplified to an equation that has the same variational formulation as the one treated in the first part of this document. Numerical simulations are done in this realistic configuration, and the different methods of resolution are compared.
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Submitted on : Wednesday, March 30, 2016 - 1:17:42 PM
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Juliette Chabassier, Marc Durufle. High Order Finite Element Method for solving Convected Helmholtz equation in radial and axisymmetric domains. Application to Helioseismology. [Research Report] RR-8893, Inria Bordeaux Sud-Ouest. 2016. ⟨hal-01295077⟩

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