A Trefftz method with reconstruction of the normal derivative applied to elliptic equations

Després, Bruno; El Ghaoui, Maria; Sayah, Toni

doi:10.1090/mcom/3756

A Trefftz method with reconstruction of the normal derivative applied to elliptic equations
HTML articles powered by AMS MathViewer

by Bruno Després, Maria El Ghaoui and Toni Sayah;

Math. Comp. 91 (2022), 2645-2679

DOI: https://doi.org/10.1090/mcom/3756

Published electronically: July 15, 2022

HTML | PDF | Request permission

Abstract:

This article deals with the application of the Trefftz method to the Laplace problem. We introduce a new discrete variational formulation using a penalisation of the continuity of the solution on the edges which is compatible with the discontinuity of the Trefftz basis functions in the cells. We prove the existence and uniqueness of the discrete solution. A high order error estimate is established. The theory is validated with several numerical experiments for different values of the mesh size, the order of the method and the penalisation coefficient. It is found that the penalisation coefficient has an influence on the conditioning of the method.

References

Douglas N. Arnold, An interior penalty finite element method with discontinuous elements, SIAM J. Numer. Anal. 19 (1982), no. 4, 742–760. MR 664882, DOI 10.1137/0719052
H. Barucq, A. Bendali, M. Fares, V. Mattesi, and S. Tordeux, A symmetric Trefftz-DG formulation based on a local boundary element method for the solution of the Helmholtz equation, J. Comput. Phys. 330 (2017), 1069–1092. MR 3581507, DOI 10.1016/j.jcp.2016.09.062
Susanne C. Brenner and L. Ridgway Scott, The mathematical theory of finite element methods, 2nd ed., Texts in Applied Mathematics, vol. 15, Springer-Verlag, New York, 2002. MR 1894376, DOI 10.1007/978-1-4757-3658-8
Haim Brezis, Functional analysis, Sobolev spaces and partial differential equations, Universitext, Springer, New York, 2011. MR 2759829, DOI 10.1007/978-0-387-70914-7
Ivo Babuška, The finite element method with penalty, Math. Comp. 27 (1973), 221–228. MR 351118, DOI 10.1090/S0025-5718-1973-0351118-5
Garth A. Baker, Finite element methods for elliptic equations using nonconforming elements, Math. Comp. 31 (1977), no. 137, 45–59. MR 431742, DOI 10.1090/S0025-5718-1977-0431742-5
Christophe Buet, Bruno Despres, and Guillaume Morel, Trefftz discontinuous Galerkin basis functions for a class of Friedrichs systems coming from linear transport, Adv. Comput. Math. 46 (2020), no. 3, Paper No. 41, 27. MR 4091990, DOI 10.1007/s10444-020-09755-5
Olivier Cessenat and Bruno Despres, Application of an ultra weak variational formulation of elliptic PDEs to the two-dimensional Helmholtz problem, SIAM J. Numer. Anal. 35 (1998), no. 1, 255–299. MR 1618464, DOI 10.1137/S0036142995285873
Olivier Cessenat and Bruno Després, Using plane waves as base functions for solving time harmonic equations with the ultra weak variational formulation, J. Comput. Acoust. 11 (2003), no. 2, 227–238. Medium-frequency acoustics. MR 2013687, DOI 10.1142/S0218396X03001912
A. Ern and J.-L. Guermond, Discontinuous Galerkin methods for Friedrichs’ systems. I. General theory, SIAM J. Numer. Anal. 44 (2006), no. 2, 753–778. MR 2218968, DOI 10.1137/050624133
Robert Eymard, Thierry Gallouët, and Raphaèle Herbin, Finite volume methods, Handbook of numerical analysis, Vol. VII, Handb. Numer. Anal., VII, North-Holland, Amsterdam, 2000, pp. 713–1020. MR 1804748, DOI 10.1016/S1570-8659(00)07005-8
Claude J. Gittelson, Ralf Hiptmair, and Ilaria Perugia, Plane wave discontinuous Galerkin methods: analysis of the $h$-version, M2AN Math. Model. Numer. Anal. 43 (2009), no. 2, 297–331. MR 2512498, DOI 10.1051/m2an/2009002
Clint Dawson, Godunov-mixed methods for advection-diffusion equations in multidimensions, SIAM J. Numer. Anal. 30 (1993), no. 5, 1315–1332. MR 1239823, DOI 10.1137/0730068
Clint Dawson, Analysis of an upwind-mixed finite element method for nonlinear contaminant transport equations, SIAM J. Numer. Anal. 35 (1998), no. 5, 1709–1724. MR 1640005, DOI 10.1137/S0036142993259421
Jim Douglas Jr. and Todd Dupont, Interior penalty procedures for elliptic and parabolic Galerkin methods, Computing methods in applied sciences (Second Internat. Sympos., Versailles, 1975) Lecture Notes in Phys., Vol. 58, Springer, Berlin-New York, 1976, pp. 207–216. MR 440955
Ivo Babuška and Miloš Zlámal, Nonconforming elements in the finite element method with penalty, SIAM J. Numer. Anal. 10 (1973), 863–875. MR 345432, DOI 10.1137/0710071
R. Herbin, Analyse numérique des équations aux dérivées partielles, Engineering School, Marseille, 2011, cel-00637008.
Jan S. Hesthaven and Tim Warburton, Nodal discontinuous Galerkin methods, Texts in Applied Mathematics, vol. 54, Springer, New York, 2008. Algorithms, analysis, and applications. MR 2372235, DOI 10.1007/978-0-387-72067-8
Ralf Hiptmair, Andrea Moiola, and Ilaria Perugia, A survey of Trefftz methods for the Helmholtz equation, Building bridges: connections and challenges in modern approaches to numerical partial differential equations, Lect. Notes Comput. Sci. Eng., vol. 114, Springer, [Cham], 2016, pp. 237–278. MR 3585792
Lise-Marie Imbert-Gérard and Peter Monk, Numerical simulation of wave propagation in inhomogeneous media using generalized plane waves, ESAIM Math. Model. Numer. Anal. 51 (2017), no. 4, 1387–1406. MR 3702418, DOI 10.1051/m2an/2016067
Tomi Huttunen, Peter Monk, and Jari P. Kaipio, Computational aspects of the ultra-weak variational formulation, J. Comput. Phys. 182 (2002), no. 1, 27–46. MR 1936802, DOI 10.1006/jcph.2002.7148
Tomi Huttunen and Peter Monk, The use of plane waves to approximate wave propagation in anisotropic media, J. Comput. Math. 25 (2007), no. 3, 350–367. MR 2320239
Jan Adam Kołodziej and Jakub Krzysztof Grabski, Many names of the Trefftz method, Eng. Anal. Bound. Elem. 96 (2018), 169–178. MR 3857577, DOI 10.1016/j.enganabound.2018.08.013
J. Nitsche, Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind, Abh. Math. Sem. Univ. Hamburg 36 (1971), 9–15 (German). MR 341903, DOI 10.1007/BF02995904
L. Hożejowsli, Trefftz method for polynomial-based boundary identification in two-dimension Laplacian problems, 2015.
Lise-Marie Imbert-Gérard and Bruno Després, A generalized plane-wave numerical method for smooth nonconstant coefficients, IMA J. Numer. Anal. 34 (2014), no. 3, 1072–1103. MR 3232445, DOI 10.1093/imanum/drt030
Boško S. Jovanović and Endre Süli, Analysis of finite difference schemes, Springer Series in Computational Mathematics, vol. 46, Springer, London, 2014. For linear partial differential equations with generalized solutions. MR 3136501, DOI 10.1007/978-1-4471-5460-0
E. Kitaa, Y. Ikedab, and N. Kamiyac, Trefftz solution for boundary value problem of three-dimensional Poisson equation, Eng. Anal. Bound. Elem. 29, 2005, 383–390.
Fritz Kretzschmar, Andrea Moiola, Ilaria Perugia, and Sascha M. Schnepp, A priori error analysis of space-time Trefftz discontinuous Galerkin methods for wave problems, IMA J. Numer. Anal. 36 (2016), no. 4, 1599–1635. MR 3556398, DOI 10.1093/imanum/drv064
E. A. W. Maunder, Trefftz in translation, Comput. Assisted Mech. and Engrg. Sci. (2003).
Guillaume Morel, Christophe Buet, and Bruno Despres, Trefftz discontinuous Galerkin method for Friedrichs systems with linear relaxation: application to the $P_1$ model, Comput. Methods Appl. Math. 18 (2018), no. 3, 521–557. MR 3824777, DOI 10.1515/cmam-2018-0006
Michelle Schatzman, Analyse numérique, InterEditions, Paris, 1991 (French, with French summary). Cours et exercices pour la licence. [Course and exercises for the bachelor’s degree]. MR 1200897
E. Trefftz, Ein Gegenstuck zum Ritzschen Verfahren, Proceedings of the 2nd International Congress for Applied Mechanics, Zurich, 1926, pp. 131-137.
O. C. Zienkiewicz, The finite element method in engineering science, McGraw-Hill, London-New York-Düsseldorf, 1971. The second, expanded and revised, edition of The finite element method in structural and continuum mechanics. MR 315970
Mary Fanett Wheeler, An elliptic collocation-finite element method with interior penalties, SIAM J. Numer. Anal. 15 (1978), no. 1, 152–161. MR 471383, DOI 10.1137/0715010