Finite element methods for parabolic equations
Author:
Miloš Zlámal
Journal:
Math. Comp. 28 (1974), 393404
MSC:
Primary 65N35
MathSciNet review:
0388813
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Abstract: The initialboundary 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 finitedifference discretizations in time: the backward, the CrankNicolson and the Calahan discretization. New error bounds are derived.
 [1]
J. H. Bramble & V. Thomée, "Semidiscreteleast squares methods for a parabolic boundary value problem," Math. Comp., v. 26, 1972, pp. 633648. MR 0349038 (50:1532)
 [2]
J. H. Bramble & V. Thomée, "Discrete time Galerkin methods for a parabolic boundary value problem," Ann. Mat. Pura Appl. (To appear.) MR 0388805 (52:9639)
 [3]
J. Douglas, Jr. & T. Dupont, "Galerkin methods for parabolic equations," SIAM J. Numer. Anal., v. 7, 1970, pp. 575626. MR 43 #2863. MR 0277126 (43:2863)
 [4]
C. W. Gear, Numerical Initial Value Problems in Ordinary Differential Equations, PrenticeHall, Englewood Cliffs, N.J., 1971. MR 0315898 (47:4447)
 [5]
J. Hlaváček, "On a semivariational method for parabolic equations. I, II," Apl. Mat., v. 17, 1972, pp. 327351; ibid., v. 18, 1973, pp. 4364. MR 0314285 (47:2837)
 [6]
O. A. Ladyženskaja, V. A. Solonnikov & N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, "Nauka", Moscow, 1967; English transl., Transl. Math. Monographs, vol. 23, Amer. Math. Soc., Providence, R.I., 1968. MR 39 #3159a,b. MR 0241822 (39:3159b)
 [7]
W. Visser, A Finite Element Method For the Determination of NonStationary Temperature Distribution and Thermal Deformations, Proc. Conf. Matrix Meth. Struct. Mech., Air Force Inst. of Techn., WrightPatterson 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. 276286.
 [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. 5981.
 [10]
M. Zlámal, "The finite element method in domains with curved boundaries," Int. J. Numer. Meth. Eng., v. 5, 1973, pp. 367373.
 [11]
M. Zlámal, "Curved elements in the finite element method. I," SIAM J. Numer. Anal., v. 10, 1973, pp. 229240. MR 0395263 (52:16060)
 [12]
M. Zlámal, "Curved elements in the finite element method. II," SIAM J. Numer. Anal., v. 11, 1974, pp. 347362. MR 0343660 (49:8400)
 [1]
 J. H. Bramble & V. Thomée, "Semidiscreteleast squares methods for a parabolic boundary value problem," Math. Comp., v. 26, 1972, pp. 633648. MR 0349038 (50:1532)
 [2]
 J. H. Bramble & V. Thomée, "Discrete time Galerkin methods for a parabolic boundary value problem," Ann. Mat. Pura Appl. (To appear.) MR 0388805 (52:9639)
 [3]
 J. Douglas, Jr. & T. Dupont, "Galerkin methods for parabolic equations," SIAM J. Numer. Anal., v. 7, 1970, pp. 575626. MR 43 #2863. MR 0277126 (43:2863)
 [4]
 C. W. Gear, Numerical Initial Value Problems in Ordinary Differential Equations, PrenticeHall, Englewood Cliffs, N.J., 1971. MR 0315898 (47:4447)
 [5]
 J. Hlaváček, "On a semivariational method for parabolic equations. I, II," Apl. Mat., v. 17, 1972, pp. 327351; ibid., v. 18, 1973, pp. 4364. MR 0314285 (47:2837)
 [6]
 O. A. Ladyženskaja, V. A. Solonnikov & N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, "Nauka", Moscow, 1967; English transl., Transl. Math. Monographs, vol. 23, Amer. Math. Soc., Providence, R.I., 1968. MR 39 #3159a,b. MR 0241822 (39:3159b)
 [7]
 W. Visser, A Finite Element Method For the Determination of NonStationary Temperature Distribution and Thermal Deformations, Proc. Conf. Matrix Meth. Struct. Mech., Air Force Inst. of Techn., WrightPatterson 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. 276286.
 [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. 5981.
 [10]
 M. Zlámal, "The finite element method in domains with curved boundaries," Int. J. Numer. Meth. Eng., v. 5, 1973, pp. 367373.
 [11]
 M. Zlámal, "Curved elements in the finite element method. I," SIAM J. Numer. Anal., v. 10, 1973, pp. 229240. MR 0395263 (52:16060)
 [12]
 M. Zlámal, "Curved elements in the finite element method. II," SIAM J. Numer. Anal., v. 11, 1974, pp. 347362. MR 0343660 (49:8400)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718197403888139
PII:
S 00255718(1974)03888139
Article copyright:
© Copyright 1974
American Mathematical Society
