We provide a comprehensive study of arbitrarily high-order finite elements defined on pyramids. We propose a new family of high-order nodal pyramidal finite element which can be used in hybrid meshes which include hexahedra, tetrahedra, wedges and pyramids. Finite elements matrices can be evaluated through approximate integration, and we show that the order of convergence of the method is conserved. Numerical results demonstrate the efficiency of hybrid meshes compared to pure tetrahedral meshes or hexahedral meshes obtained by splitting tetrahedra into hexahedra.

Context. Local helioseismology has so far relied on semi-analytical methods to compute the spatial sensitivity of wave travel times to perturbations in the solar interior. These methods are cumbersome and lack flexibility. Aims. Here we propose a convenient framework for numerically solving the forward problem of time-distance helioseismology in the frequency domain. The fundamental quantity to be computed is the cross-covariance of the seismic wavefield. Methods. We choose sources of wave excitation that enable us to relate the cross-covariance of the oscillations to the Green's function in a straightforward manner. We illustrate the method by considering the 3D acoustic wave equation in an axisymmetric reference solar model, ignoring the effects of gravity on the waves. The symmetry of the background model around the rotation axis implies that the Green's function can be written as a sum of longitudinal Fourier modes, leading to a set of independent 2D problems. We use a high-order finite-element method to solve the 2D wave equation in frequency space. The computation is 'embarrassingly parallel', with each frequency and each azimuthal order solved independently on a computer cluster. Results. We compute travel-time sensitivity kernels in spherical geometry for flows, sound speed, and density perturbations under the first Born approximation. Convergence tests show that travel times can be computed with a numerical precision better than one millisecond, as required by the most precise travel-time measurements. Conclusions. The method presented here is computationally efficient and will be used to interpret travel-time measurements in order to infer, e.g., the large-scale meridional flow in the solar convection zone. It allows the implementation of (full-waveform) iterative inversions, whereby the axisymmetric background model is updated at each iteration.

RésuméNous effectuons une revue succincte de l'obtention et l'utilisation des conditions d'impédances généralisées (GIBC) dans le cas de revêtements diélectriques minces et dans le cas d'objets fortement conducteurs. Nous nous plaçons dans le cadre des probèmes de diffraction d'ondesélectromagnétiques en régime harmonique. Nous testons numériquement la validité et la précision de ces conditions aux limites pour le cas de forte conductivité. En présence de géométries comportant des coins, nous proposons un traitement numérique astucieux afin de garder la même précision que dans le cas de géométries régulières. AbstractWe briefly review the use and the derivation of Generalized Impedance Boundary Conditions (GIBC) in the case of thin dielectric coatings and in the case of strongly absorbing media, within the context of electromagnetic scattering problem at a fixed frequency. We then numerically test the validity and accuracy of these boundary conditions in the case of high absorption. A numerical treatment of the corner singularity is proposed to recover the accuracy of the GIBC for singular geometries.

Abstract. This work offers some contributions to the numerical study of acoustic waves propagating in the Sun and its atmosphere. The main goal is to provide boundary conditions for outgoing waves in the solar atmosphere where it is assumed that the sound speed is constant and the density decays exponentially with radius. Outgoing waves are governed by a Dirichlet-to-Neumann map which is obtained from the factorization of the Helmholtz equation expressed in spherical coordinates. For the purpose of extending the outgoing wave equation to axisymmetric or 3D cases, different approximations are implemented by using the frequency and/or the angle of incidence as parameters of interest. This results in boundary conditions called Atmospheric Radiation Boundary Conditions (ARBC) which are tested in ideal and realistic configurations. These ARBCs deliver accurate results and reduce the computational burden by a factor of two in helioseismology applications.Résumé. Ce travail apporte quelques contributions à l'étude numérique des ondes acoustiques se propageant dans le Soleil et son atmosphère. Il se base sur la caractérisa-tion des ondes sortantes dans l'atmosphère représentée par une vitesse constante et une densité décroissant exponentiellement. Les ondes sortantes sont régies par un opérateur Dirichlet-to-Neumann qui est obtenu par la factorisation de l'équation de Helmholtz formulée dans les coordonnées sphériques. Afin d'étendre l'équation des ondes sortantes à des géométries axisymétriques ou 3D, différentes approximations sont menées en utilisant la fréquence et/ou l'angle d'incidence comme paramètres d'intérêt. Ceci mène à des conditions de frontière que nous appelons Conditions de Radiation Atmosphériques (ARBC) et qui sont testées en configuration idéalisées et réalistes. Ces conditions ARBC offrent des résultats précis et réduisent le coût de calcul d'un facteur deux pour le cas du Soleil.1991 Mathematics Subject Classification. 00A71, 35L05, 85A20, 33C55, 65M60.December 1, 2017.