Conference Agenda

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Session Overview
Session
MS 22a: high-order face-based discretization methods
Time:
Monday, 12/July/2021:
1:50pm - 3:50pm

Session Chair: Théophile Chaumont-Frelet
Session Chair: Simon Lemaire
Session Chair: Alexandre Ern
Virtual location: Zoom 5


Session Abstract

This minisymposium focuses on face-based discretization methods, i.e. Galerkin methods

with unknowns attached to the mesh faces. These methods present several advantages,

including (i) their built-in static condensation properties leading to improved performances,

(ii) their native possibility to handle high polynomial degrees of approximation, (iii) their

natural ability to preserve at the discrete level key physical features of the continuous

model (local balances, limit behaviors), as well as (iv) their capability to handle general

polytopal meshes (hence simplifying adaptive refinement/coarsening).

Examples of face-based discretization methods encompass (but are not limited to) the Hy-

bridizable Discontinuous Galerkin (HDG) method, the Hybrid High-Order (HHO) method,

or the Multiscale Hybrid-Mixed (MHM) method.

The objective of this minisymposium is twofold. On the one hand, state-of-the-art advances

in face-based discretization methods will be presented, with applications to various areas

of mechanics or physics. On the other hand, this minisymposium aims at highlighting the

connections and at building bridges between existing face-based discretization methods

and the associated communities.


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Presentations
1:50pm - 2:20pm

Convergence of an HDG finite element method for Maxwell's equations in an inhomogeneous medium

Peter Monk1, Gang Chen2, Yangwen Zhang1

1University of Delaware, United States of America; 2Sichuan University, China

We analyze the convergence of a hybridizable discontinuous Galerkin (HDG) method for Maxwell's equations. Our method uses standard HDG spaces to discretize the electromagnetic fields, and a continuous space to discretize the Lagrange multiplier for the divergence constraint. Our goal is to prove convergence results for a general heterogeneous medium. We adapt the techniques of Buffa and Perugia to the HDG setting, and prove convergence of a piecewise smooth medium. Numerical experiments confirm the theoretical results.



2:20pm - 2:50pm

Hybrid High-Order methods for three-dimensional magnetostatics

Simon Lemaire

INRIA, Lille, France

We discuss the application of the Hybrid High-Order (HHO) technology to the arbitrary-order approximation of magnetostatics on general (three-dimensional) polyhedral meshes. We consider both the (first-order) field formulation of magnetostatics and the (second-order) vector potential formulation of the problem. For these two formulations, we devise and analyze HHO methods. The well-posedness of these methods hinges on a discrete version of (the first) Weber inequality on hybrid (cell- and face-based) spaces. We illustrate our findings on a set of test-cases, and we discuss the links between our methods and the HDG literature. This is a joint work with Florent Chave (Univ. Côte d'Azur) and Daniele A. Di Pietro (Univ. Montpellier).



2:50pm - 3:20pm

Convergence analysis of HDG formulations for the convected Helmholtz equation

Nathan Rouxelin2,1, Hélène Barucq1,2, Sébastien Tordeux2,1

1Team Makutu, Inria Bordeaux Sud-Ouest, France; 2LMAP, e2s-UPPA, CNRS, Université de Pau et des Pays de l'Adour, France

Our objective is to develop a particular branch for the simulation of aeroacoustic problems suitable for helioseismic numerical studies in the open-source software Hawen. Given that Hybridizable Discontinuous Galerkin (HDG) methods have demonstrated their efficiency for solving seismic inverse problems, we propose to build our computational environment upon HDG formulations.

As a first step toward the development of efficient and reliable numerical methods for computational helioseismology, we present three variations of the HDG method for the convected Helmholtz equation, which is the most simple aeroacoustic model. HDG methods are mixed DG methods in which a new unknown that only lives on the skeleton of mesh is introduced. This allows an efficient implementation of the method that relies on a static condensation process, leading to a so-called global problem for this skeleton unknown only. The original unknowns can then be reconstructed locally in a parallel way. HDG methods therefore keep the advantages of the DG methods (eg. hp-adaptivity, high order,…) without the high numerical cost of those methods due to the duplication of the degrees of freedom associated with the interfaces of the elements.

For these three HDG methods, we present theoretical results including well-posedness of the local and global problems, convergence analysis for regular solutions, and a discussion on the choice of penalization parameters. We also present implementation details and numerical experiments to illustrate the theoretical results, as well as illustrative examples to demonstrate the possibilities of the method in more realistic situations.



 
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