A Galerkin method for the forward-backward heat equation
HTML articles powered by AMS MathViewer
- by A. K. Aziz and J.-L. Liu PDF
- Math. Comp. 56 (1991), 35-44 Request permission
Abstract:
In this paper a new variational method is proposed for the numerical approximation of the solution of the forward-backward heat equation. The approach consists of first reducing the second-order problem to an equivalent first-order system, and then using a finite element procedure with continuous elements in both space and time for the numerical approximation. Under suitable regularity assumptions, error estimates and the results of some numerical experiments are presented.References
- A. K. Aziz and Peter Monk, Continuous finite elements in space and time for the heat equation, Math. Comp. 52 (1989), no. 186, 255–274. MR 983310, DOI 10.1090/S0025-5718-1989-0983310-2
- A. K. Aziz, S. Leventhal, and A. Werschulz, Higher-order convergence for a finite element method for the Tricomi problem, Numer. Funct. Anal. Optim. 2 (1980), no. 1, 65–78. MR 580383, DOI 10.1080/01630568008816045
- M. S. Baouendi and P. Grisvard, Sur une équation d’évolution changeant de type, J. Functional Analysis 2 (1968), 352–367 (French). MR 0252817, DOI 10.1016/0022-1236(68)90012-8
- K. O. Friedrichs, Symmetric positive linear differential equations, Comm. Pure Appl. Math. 11 (1958), 333–418. MR 100718, DOI 10.1002/cpa.3160110306 M. Gevrey, Sur les équations aux dérivées partielles du type parabolique, J. Math. Pures Appl. (6) 9 (1913), 305-475. —, Sur les équations aux dérivées partielles du type parabolique (suite), J. Math. Pures Appl. (6) 10 (1914), 105-148.
- Jerome A. Goldstein and Tapas Mazumdar, A heat equation in which the diffusion coefficient changes sign, J. Math. Anal. Appl. 103 (1984), no. 2, 533–564. MR 762573, DOI 10.1016/0022-247X(84)90145-8 T. LaRosa, The propagation of an electron beam through the solar corona, Ph.D. Dissertation, Department of Physics and Astronomy, University of Maryland, 1986.
- P. Lesaint and P.-A. Raviart, Finite element collocation methods for first-order systems, Math. Comp. 33 (1979), no. 147, 891–918. MR 528046, DOI 10.1090/S0025-5718-1979-0528046-0 J. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod Gauthier-Villars, Paris, 1969. K. Stewartson, Multistructural boundary layers on flat plates and related bodies, Adv. in Appl. Mech. 14 (1974), 145-239.
- Keith Stewartson, d’Alembert’s paradox, SIAM Rev. 23 (1981), no. 3, 308–343. MR 631832, DOI 10.1137/1023063 V. Thomée, Galerkin finite element methods for parabolic problems, Lecture Notes in Math., vol. 1054, Springer-Verlag, 1972. V. Vanaja, Iterative solutions of backward-forward heat equation, Ph.D. Dissertation, Department of Mathematics, University of Maryland, 1988.
Additional Information
- © Copyright 1991 American Mathematical Society
- Journal: Math. Comp. 56 (1991), 35-44
- MSC: Primary 65M60
- DOI: https://doi.org/10.1090/S0025-5718-1991-1052085-2
- MathSciNet review: 1052085