Finite element collocation methods for first-order systems

Lesaint, P.; Raviart, P.-A.

doi:10.1090/S0025-5718-1979-0528046-0

Finite element collocation methods for first-order systems
HTML articles powered by AMS MathViewer

by P. Lesaint and P.-A. Raviart PDF

Math. Comp. 33 (1979), 891-918 Request permission

Abstract:

Finite element methods and the associate collocation methods are considered for solving first-order hyperbolic systems, positive in the sense of Friedrichs. Applied in the case when the meshes are rectangle, those methods lead for example to the successfully used box scheme for the heat equation or D.S.N. scheme for the neutron transport equation. Generalizations of these methods are described here for nonrectangle meshes and (or) noncylindrical domains; stability results and error estimates are derived.

References

Garth A. Baker, A finite element method for first order hyperbolic equations, Math. Comput. 29 (1975), no. 132, 995–1006. MR 0400744, DOI 10.1090/S0025-5718-1975-0400744-5
P. G. Ciarlet and P.-A. Raviart, General Lagrange and Hermite interpolation in $\textbf {R}^{n}$ with applications to finite element methods, Arch. Rational Mech. Anal. 46 (1972), 177–199. MR 336957, DOI 10.1007/BF00252458
P. G. Ciarlet and P.-A. Raviart, Interpolation theory over curved elements, with applications to finite element methods, Comput. Methods Appl. Mech. Engrg. 1 (1972), 217–249. MR 375801, DOI 10.1016/0045-7825(72)90006-0
K. O. Friedrichs, Symmetric positive linear differential equations, Comm. Pure Appl. Math. 11 (1958), 333–418. MR 100718, DOI 10.1002/cpa.3160110306
Bernie L. Hulme, Discrete Galerkin and related one-step methods for ordinary differential equations, Math. Comp. 26 (1972), 881–891. MR 315899, DOI 10.1090/S0025-5718-1972-0315899-8
Bernie L. Hulme, One-step piecewise polynomial Galerkin methods for initial value problems, Math. Comp. 26 (1972), 415–426. MR 321301, DOI 10.1090/S0025-5718-1972-0321301-2
Herbert B. Keller, A new difference scheme for parabolic problems, Numerical Solution of Partial Differential Equations, II (SYNSPADE 1970) (Proc. Sympos., Univ. of Maryland, College Park, Md., 1970) Academic Press, New York, 1971, pp. 327–350. MR 0277129

Computing Methods in Reactor Physics

Sur la Résolution des Systèmes Hyperboliques du Premier Ordre par des Méthodes d’Éléments Finis

P. Lesaint, Finite element methods for symmetric hyperbolic equations, Numer. Math. 21 (1973/74), 244–255. MR 341902, DOI 10.1007/BF01436628
P. Lesaint, Finite element methods for the transport equation, Rev. Française Automat. Informat. Recherche Opérationnelle Ser. Rouge 8 (1974), no. R-2, 67–93 (English, with Loose French summary). MR 0408677, DOI 10.1016/0010-4485(73)90015-8
P. Lasaint and P.-A. Raviart, On a finite element method for solving the neutron transport equation, Mathematical aspects of finite elements in partial differential equations (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1974) Publication No. 33, Math. Res. Center, Univ. of Wisconsin-Madison, Academic Press, New York, 1974, pp. 89–123. MR 0658142
Niel K. Madsen, Convergent centered difference schemes for the discrete ordinate neutron transport equations, SIAM J. Numer. Anal. 12 (1975), 164–176. MR 405882, DOI 10.1137/0712015
P.-A. Raviart and J. M. Thomas, Primal hybrid finite element methods for $2$nd order elliptic equations, Math. Comp. 31 (1977), no. 138, 391–413. MR 431752, DOI 10.1090/S0025-5718-1977-0431752-8