Approximation Properties of Vectorial Exponential Functions

doi:10.1007/s42967-023-00310-9

Communications on Applied Mathematics and Computation ›› 2024, Vol. 6 ›› Issue (3): 1801-1831.doi: 10.1007/s42967-023-00310-9

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Approximation Properties of Vectorial Exponential Functions

Christophe Buet^1,2, Bruno Despres³, Guillaume Morel⁴

1 CEA, DAM, DIF, 91297 Arpajon, France;
2 Laboratoire en Informatique Haute Performance Pour le Calcul et la Simulation, CEA, Université Paris-Saclay, 91680 Bruyères-le-Châtel, France;
3 Laboratoire Jacques-Louis Lions, UMR 7598, Sorbonne Université, 75005 Paris, France;
4 IMT Atlantique, 29238 Brest, France

Received:2022-10-22 Revised:2023-08-14 Accepted:2023-08-16 Published:2024-12-20
Contact: Bruno Despres,bruno.despres@sorbonne-universite.fr;Christophe Buet,christophe.buet@cea.fr;Guillaume Morel,guillaume.morel@imt-atlantique.fr E-mail:bruno.despres@sorbonne-universite.fr;christophe.buet@cea.fr;guillaume.morel@imt-atlantique.fr

Abstract

Abstract: This contribution is dedicated to the celebration of Rémi Abgrall’s accomplishments in Applied Mathematics and Scientific Computing during the conference “Essentially Hyperbolic Problems: Unconventional Numerics, and Applications”. With respect to classical Finite Elements Methods, Trefftz methods are unconventional methods because of the way the basis functions are generated. Trefftz discontinuous Galerkin (TDG) methods have recently shown potential for numerical approximation of transport equations [6, 26] with vectorial exponential modes. This paper focuses on a proof of the approximation properties of these exponential solutions. We show that vectorial exponential functions can achieve high order convergence. The fundamental part of the proof consists in proving that a certain rectangular matrix has maximal rank.

Key words: Trefftz method, Transport equation, Vectorial exponential functions

CLC Number:

Christophe Buet, Bruno Despres, Guillaume Morel. Approximation Properties of Vectorial Exponential Functions[J]. Communications on Applied Mathematics and Computation, 2024, 6(3): 1801-1831.

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Approximation Properties of Vectorial Exponential Functions

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