Finite element collocation methods for first-order systems

Authors:
P. Lesaint and P.-A. Raviart

Journal:
Math. Comp. **33** (1979), 891-918

MSC:
Primary 65N30

DOI:
https://doi.org/10.1090/S0025-5718-1979-0528046-0

MathSciNet review:
528046

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

**[1]**Garth A. Baker,*A finite element method for first order hyperbolic equations*, Math. Comput.**29**(1975), no. 132, 995–1006. MR**0400744**, https://doi.org/10.1090/S0025-5718-1975-0400744-5**[2]**P. G. Ciarlet and P.-A. Raviart,*General Lagrange and Hermite interpolation in 𝑅ⁿ with applications to finite element methods*, Arch. Rational Mech. Anal.**46**(1972), 177–199. MR**0336957**, https://doi.org/10.1007/BF00252458**[3]**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**0375801**, https://doi.org/10.1016/0045-7825(72)90006-0**[4]**K. O. Friedrichs,*Symmetric positive linear differential equations*, Comm. Pure Appl. Math.**11**(1958), 333–418. MR**0100718**, https://doi.org/10.1002/cpa.3160110306**[5]**Bernie L. Hulme,*Discrete Galerkin and related one-step methods for ordinary differential equations*, Math. Comp.**26**(1972), 881–891. MR**0315899**, https://doi.org/10.1090/S0025-5718-1972-0315899-8**[6]**Bernie L. Hulme,*One-step piecewise polynomial Galerkin methods for initial value problems*, Math. Comp.**26**(1972), 415–426. MR**0321301**, https://doi.org/10.1090/S0025-5718-1972-0321301-2**[7]**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****[8]**K. D. LATHROP & B. G. CARLSON, "Transport theory. The method of discrete ordinates,"*Computing Methods in Reactor Physics*(Greenspan, Kelerb, Okrent, Eds.), Gordon and Breach, New York, 1968, pp. 165-266.**[9]**P. LESAINT,*Sur la Résolution des Systèmes Hyperboliques du Premier Ordre par des Méthodes d'Éléments Finis*, Doctoral thesis, Paris, 1975.**[10]**P. Lesaint,*Finite element methods for symmetric hyperbolic equations*, Numer. Math.**21**(1973/74), 244–255. MR**0341902**, https://doi.org/10.1007/BF01436628**[11]**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****[12]**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), Math. Res. Center, Univ. of Wisconsin-Madison, Academic Press, New York, 1974, pp. 89–123. Publication No. 33. MR**0658142****[13]**Niel K. Madsen,*Convergent centered difference schemes for the discrete ordinate neutron transport equations*, SIAM J. Numer. Anal.**12**(1975), 164–176. MR**0405882**, https://doi.org/10.1137/0712015**[14]**P.-A. Raviart and J. M. Thomas,*Primal hybrid finite element methods for 2nd order elliptic equations*, Math. Comp.**31**(1977), no. 138, 391–413. MR**0431752**, https://doi.org/10.1090/S0025-5718-1977-0431752-8

Retrieve articles in *Mathematics of Computation*
with MSC:
65N30

Retrieve articles in all journals with MSC: 65N30

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1979-0528046-0

Article copyright:
© Copyright 1979
American Mathematical Society