Finite element methods for parabolic equations

Author:
Miloš Zlámal

Journal:
Math. Comp. **28** (1974), 393-404

MSC:
Primary 65N35

DOI:
https://doi.org/10.1090/S0025-5718-1974-0388813-9

MathSciNet review:
0388813

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Abstract | References | Similar Articles | Additional Information

Abstract: The initial-boundary value problem for a linear parabolic equation with the Dirichlet boundary condition is solved approximately by applying the finite element discretization in the space dimension and three types of finite-difference discretizations in time: the backward, the Crank-Nicolson and the Calahan discretization. New error bounds are derived.

**[1]**James H. Bramble and Vidar Thomée,*Semidiscrete least-squares methods for a parabolic boundary value problem*, Math. Comp.**26**(1972), 633–648. MR**349038**, https://doi.org/10.1090/S0025-5718-1972-0349038-4**[2]**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**, https://doi.org/10.1007/BF02417101**[3]**Jim Douglas Jr. and Todd Dupont,*Galerkin methods for parabolic equations*, SIAM J. Numer. Anal.**7**(1970), 575–626. MR**277126**, https://doi.org/10.1137/0707048**[4]**C. William Gear,*Numerical initial value problems in ordinary differential equations*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1971. MR**0315898****[5]**Ivan Hlaváček,*On a semi-variational method for parabolic equations. I*, Apl. Mat.**17**(1972), 327–351 (English, with Czech summary). MR**314285****[6]**O. A. Ladyženskaja, V. A. Solonnikov, and N. N. Ural′ceva,*Lineĭ nye i kvazilineĭ nye uravneniya parabolicheskogo tipa*, Izdat. “Nauka”, Moscow, 1967 (Russian). MR**0241821****[7]**W. Visser,*A Finite Element Method For the Determination of Non-Stationary Temperature Distribution and Thermal Deformations*, Proc. Conf. Matrix Meth. Struct. Mech., Air Force Inst. of Techn., Wright-Patterson A. F. Base, Ohio, 1965.**[8]**E. L. Wilson & R. E. Nickell, "Application of finite element method to heat conduction analysis,"*Nuclear Eng. Design*, v. 4, 1966, pp. 276-286.**[9]**M. Zlámal, "Some recent advances in the mathematics of finite elements," in*The Mathematics of Finite Elements and Applications*, edited by J. R. Whiteman, Academic Press, London, 1972, pp. 59-81.**[10]**M. Zlámal, "The finite element method in domains with curved boundaries,"*Int. J. Numer. Meth. Eng.*, v. 5, 1973, pp. 367-373.**[11]**Miloš Zlámal,*Curved elements in the finite element method. I*, SIAM J. Numer. Anal.**10**(1973), 229–240. MR**395263**, https://doi.org/10.1137/0710022**[12]**Miloš Zlámal,*Curved elements in the finite element method. II*, SIAM J. Numer. Anal.**11**(1974), 347–362. MR**343660**, https://doi.org/10.1137/0711031

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DOI:
https://doi.org/10.1090/S0025-5718-1974-0388813-9

Article copyright:
© Copyright 1974
American Mathematical Society