Finite element multistep discretizations of parabolic boundary value problems

Zlámal, Miloš

doi:10.1090/S0025-5718-1975-0371105-2

Finite element multistep discretizations of parabolic boundary value problems
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Math. Comp. 29 (1975), 350-359 Request permission

Abstract:

The initial-boundary value problem for a linear parabolic equation in an infinite cylinder under the Dirichlet boundary condition is solved by applying the finite element discretization in the space dimension and ${A_0}$-stable multistep discretizations in time. No restriction on the ratio of the time and space increments is imposed. The methods are analyzed and bounds for the discretization error in the ${L_2}$-norm are given.

References

James H. Bramble and Vidar Thomée, Discrete time Galerkin methods for a parabolic boundary value problem, Ann. Mat. Pura Appl. (4) 101 (1974), 115–152. MR 388805, DOI 10.1007/BF02417101
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
Colin W. Cryer, A new class of highly-stable methods: $A_{0}$-stable methods, Nordisk Tidskr. Informationsbehandling (BIT) 13 (1973), 153–159. MR 323111, DOI 10.1007/bf01933487
Colin W. Cryer, On the instability of high order backward-difference multistep methods, Nordisk Tidskr. Informationsbehandling (BIT) 12 (1972), 17–25. MR 311112, DOI 10.1007/bf01932670
Peter Henrici, Discrete variable methods in ordinary differential equations, John Wiley & Sons, Inc., New York-London, 1962. MR 0135729

Linear and Quasi-Linear Equations of Parabolic Type

39

J. D. Lambert, Computational methods in ordinary differential equations, John Wiley & Sons, London-New York-Sydney, 1973. Introductory Mathematics for Scientists and Engineers. MR 0423815
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 0315970
Miloš Zlámal, Finite element methods for parabolic equations, Math. Comp. 28 (1974), 393–404. MR 388813, DOI 10.1090/S0025-5718-1974-0388813-9
Miloš Zlámal, Curved elements in the finite element method. I, SIAM J. Numer. Anal. 10 (1973), 229–240. MR 395263, DOI 10.1137/0710022
Miloš Zlámal, Curved elements in the finite element method. II, SIAM J. Numer. Anal. 11 (1974), 347–362. MR 343660, DOI 10.1137/0711031