Continuous finite elements in space and time for the heat equation

Authors:
A. K. Aziz and Peter Monk

Journal:
Math. Comp. **52** (1989), 255-274

MSC:
Primary 65N30

DOI:
https://doi.org/10.1090/S0025-5718-1989-0983310-2

MathSciNet review:
983310

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we shall analyze a new variational method for approximating the heat equation using continuous finite elements in space and time. In the special case of linear elements in time the method reduces to the Crank-Nicolson Galerkin method with time-averaged data. Using higher-order finite elements in time, we obtain a new class of time stepping methods related to collocating the standard spatial Galerkin differential equations in time at the Gauss-Legendre points. Again the data enters via suitable time averages. We present error estimates and the results of some numerical experiments.

**[1]**Ivo Babuška and A. K. Aziz,*Survey lectures on the mathematical foundations of the finite element method*, The mathematical foundations of the finite element method with applications to partial differential equations (Proc. Sympos., Univ. Maryland, Baltimore, Md., 1972) Academic Press, New York, 1972, pp. 1–359. With the collaboration of G. Fix and R. B. Kellogg. MR**0421106****[2]**Philippe G. Ciarlet,*\cyr Metod konechnykh èlementov dlya èllipticheskikh zadach*, “Mir”, Moscow, 1980 (Russian). Translated from the English by B. I. Kvasov. MR**608971****[3]**Jim Douglas Jr. and Todd Dupont,*Collocation methods for parabolic equations in a single space variable*, Lecture Notes in Mathematics, Vol. 385, Springer-Verlag, Berlin-New York, 1974. Based on 𝐶¹-piecewise-polynomial spaces. MR**0483559****[4]**Jim Douglas Jr., Todd Dupont, and Mary F. Wheeler,*A quasi-projection analysis of Galerkin methods for parabolic and hyperbolic equations*, Math. Comp.**32**(1978), no. 142, 345–362. MR**0495012**, https://doi.org/10.1090/S0025-5718-1978-0495012-2**[5]**Kenneth Eriksson and Claes Johnson,*Error estimates and automatic time step control for nonlinear parabolic problems. I*, SIAM J. Numer. Anal.**24**(1987), no. 1, 12–23. MR**874731**, https://doi.org/10.1137/0724002**[6]**Kenneth Eriksson, Claes Johnson, and Vidar Thomée,*Time discretization of parabolic problems by the discontinuous Galerkin method*, RAIRO Modél. Math. Anal. Numér.**19**(1985), no. 4, 611–643 (English, with French summary). MR**826227**, https://doi.org/10.1051/m2an/1985190406111**[7]**Richard S. Falk and Gerard R. Richter,*Analysis of a continuous finite element method for hyperbolic equations*, SIAM J. Numer. Anal.**24**(1987), no. 2, 257–278. MR**881364**, https://doi.org/10.1137/0724021**[8]**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**[9]**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**[10]**Pierre Jamet,*Galerkin-type approximations which are discontinuous in time for parabolic equations in a variable domain*, SIAM J. Numer. Anal.**15**(1978), no. 5, 912–928. MR**507554**, https://doi.org/10.1137/0715059**[11]**Pierre Jamet,*Stability and convergence of a generalized Crank-Nicolson scheme on a variable mesh for the heat equation*, SIAM J. Numer. Anal.**17**(1980), no. 4, 530–539. MR**584728**, https://doi.org/10.1137/0717045**[12]**O. A. Ladyženskaja, V. A. Solonnikov, and N. N. Ural′ceva,*Linear and quasilinear equations of parabolic type*, Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, R.I., 1968 (Russian). MR**0241822****[13]**P. Lesaint and P.-A. Raviart,*Finite element collocation methods for first-order systems*, Math. Comp.**33**(1979), no. 147, 891–918. MR**528046**, https://doi.org/10.1090/S0025-5718-1979-0528046-0**[14]**J. L. Lions & E. Magenes,*Non-Homogeneous Boundary Problems and Applications I*, Springer-Verlag, New York, 1972.**[15]**J. L. Lions & E. Magenes,*Non-Homogeneous Boundary Problems and Applications II*, Springer-Verlag, New York, 1972.**[16]**Mitchell Luskin and Rolf Rannacher,*On the smoothing property of the Galerkin method for parabolic equations*, SIAM J. Numer. Anal.**19**(1982), no. 1, 93–113. MR**646596**, https://doi.org/10.1137/0719003**[17]**S. M. Nikol'skiI,*Approximation of Functions of Several Variables and Embedding Theorems*, Springer-Verlag, New York, 1975.**[18]**V. Thomée,*Galerkin Finite Element Methods for Parabolic Problems*, Lecture Notes in Math., Vol. 1054, Springer-Verlag, Berlin and New York, 1980.**[19]**Vidar Thomée,*Negative norm estimates and superconvergence in Galerkin methods for parabolic problems*, Math. Comp.**34**(1980), no. 149, 93–113. MR**551292**, https://doi.org/10.1090/S0025-5718-1980-0551292-5**[20]**Mary Fanett Wheeler,*A priori 𝐿₂ error estimates for Galerkin approximations to parabolic partial differential equations*, SIAM J. Numer. Anal.**10**(1973), 723–759. MR**0351124**, https://doi.org/10.1137/0710062**[21]**Ragnar Winther,*A stable finite element method for initial-boundary value problems for first-order hyperbolic systems*, Math. Comp.**36**(1981), no. 153, 65–86. MR**595042**, https://doi.org/10.1090/S0025-5718-1981-0595042-6

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/S0025-5718-1989-0983310-2

Keywords:
Galerkin method,
heat equation,
error estimates

Article copyright:
© Copyright 1989
American Mathematical Society