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

收稿日期:2022-10-22 修回日期:2023-08-14 接受日期:2023-08-16 发布日期:2024-12-20
通讯作者: 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

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

摘要/Abstract

摘要： 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.

关键词: Trefftz method, Transport equation, Vectorial exponential functions

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

中图分类号:

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

参考文献

1. Avvakumov, A.V., Strizhov, V.F., Vabishchevich, P.N., Vasilev, A.O.: Numerical modeling of neutron transport in SP3 approximation by finite element method. arXiv: 1903. 11502 v1 (2019)
2. Azmy, Y., Sartori, E.: Nuclear Computational Science: a Century in Review. Springer, Berlin (2010)
3. Bell, G., Glasstone, S.: Nuclear Reactor Theory. Van Nostrand Reinhold Company, New York (1970)
4. Brenner, S.C., Scott, L.: The Mathematical Theory of Finite Element Methods, 3rd edn. Springer, New York (2008)
5. Buet, C., Després, B., Morel, G.: Discretization of the PN model for 2D transport of particles with a Trefftz discontinuous Galerkin method (2019). https:// hal. sorbo nne- unive rsite. fr/ hal- 02372 279/ document
6. Buet, C., Despres, B., Morel, G.: Trefftz discontinuous Galerkin basis functions for a class of Friedrichs systems coming from linear transport. ACOM 4, 1–27 (2020)
7. Buffa, A., Monk, P.: Error estimates for the Ultra Weak Variational Formulation of the Helmholtz equation. ESAIM Math. Model. Numer. Anal. 42, 925–940 (2008)
8. Cessenat, O., Després, B.: Application of an ultra weak variational formulation of elliptic PDEs to the two-dimensional Helmholtz problem. SIAM J. Numer. Anal. 35, 255–299 (1998)
9. Cessenat, O., Després, B.: Using plane waves as base functions for solving time harmonic equations with the ultra weak variational formulation. Medium Freq. Acoust. 11, 227–238 (2003)
10. Crockatt, M.M., Christlieb, A.J., Garrett, C.K., Hauck, C.D.: Hybrid methods for radiation transport using diagonally implicit Runge-Kutta and space-time discontinuous Galerkin time integration. J. Comput. Phys. 376, 455–477 (2019)
11. Després, B., El Ghaoui, M., Sayah, T.: A Trefftz method with reconstruction of the normal derivative applied to elliptic equations. Math. Comp. 91, 2645–2679 (2022)
12. Gittelson, C.J., Hiptmair, R., Perugia, I.: Plane wave discontinuous Galerkin methods: analysis of the h-version. ESAIM Math. Model. Numer. Anal. 43, 297–331 (2009)
13. Heningburg, V., Hauck, C.D.: A hybrid finite-volume, discontinuous Galerkin discretization for the radiative transport equation. Multiscale Model. Simul. 19, 1–24 (2021)
14. Hermeline, F.: A discretization of the multigroup PN radiative transfer equation on general meshes. J. Comput. Phys. 313, 549–582 (2016)
15. Hiptmair, R., Moiola, A., Perugia, I.: Plane wave discontinuous Galerkin methods for the 2D Helmholtz equation: analysis of the p-version. SIAM J. Numer. Anal. 49, 264–284 (2011)
16. Hiptmair, R., Moiola, A., Perugia, I.: Plane wave discontinuous Galerkin methods: exponential convergence of the hp-version. Found. Comput. Math. 16, 637–675 (2016)
17. Huttunen, T., Monk, P., Kaipio, J.P.: Computational aspects of the ultra-weak variational formulation. J. Comput. Phys. 182, 27–46 (2002)
18. Imbert-Gérard, L.-M.: Interpolation properties of generalized plane waves. Numer. Math. 131, 683–711 (2015)
19. Imbert-Gerard, L.-M.: Amplitude-based generalized plane waves: new quasi-Trefftz functions for scalar equations in two dimensions. SIAM J. Numer. Anal. 59, 1663–1686 (2021)
20. Imbert-Gérard, L.-M., Després, B.: A generalized plane-wave numerical method for smooth nonconstant coefficients. IMA J. Numer. Anal. 34, 1072–1103 (2014)
21. Lehrenfeld, C., Stocker, P.: Embedded Trefftz discontinuous Galerkin methods (2022). arXiv: 2201. 07041
22. McClarren, R.G.: Theoretical aspects of the simplified PN equations. Transp. Theory Stat. Phys. 39, 73–109 (2010)
23. Mihalas, D., Mihalas, B.W.: Foundations of Radiation Hydrodynamics. Oxford University Press, New York (1984)
24. Moiola, A., Hiptmair, R., Perugia, I.: Plane wave approximation of homogeneous Helmholtz solutions. Z. Angew. Math. Phys. 62, 809–837 (2011)
25. Morel, G.: Asymptotic-preserving and well-balanced schemes for transport models using Trefftz discontinuous Galerkin method, theses, Sorbonne Université (2018). https:// hal. archi ves- ouver tes. fr/ tel-01911 872
26. Morel, G., Buet, C., Després, B.: Trefftz discontinuous Galerkin method for Friedrichs systems with linear relaxation: application to the P1 model. Comput. Methods Appl. Math. 18, 521–557 (2018)
27. Pomraning, G.C.: The Equations of Radiation Hydrodynamics, International Series of Monographs in Natural Philosophy. Pergamon Press, Oxford (1973)
28. Zienkiewicz, O.: Origins, milestones and directions of the finite element method—A personal view. Archives of Computational Methods in Engineering 2, 1–48 (1995)

Approximation Properties of Vectorial Exponential Functions

Approximation Properties of Vectorial Exponential Functions

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