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 Free Access

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]**I. Babuška & A. K. Aziz, "Survey lecture on the mathematical foundations of the finite element method," in*The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations*(A. K. Aziz, ed.), Academic Press, New York, 1972, pp. 5-359. MR**0421106 (54:9111)****[2]**P. G. Ciarlet,*The Finite Element Method for Elliptic Problems*, North-Holland, New York, 1980. MR**608971 (82c:65068)****[3]**J. Douglas, Jr. & T. Dupont,*Collocation Methods for Parabolic Equations in a Single Space Variable*, Lecture Notes in Math., Vol. 385, Springer-Verlag, Berlin and New York, 1974. MR**0483559 (58:3551)****[4]**J. Douglas, Jr., T. Dupont & M. F. Wheeler, "A quasi projection analysis of Galerkin methods for parabolic and hyperbolic equations,"*Math. Comp.*, v. 32, 1978, pp. 345-362. MR**0495012 (58:13780)****[5]**K. Eriksson & C. Johnson, "Error estimates and automatic time step control for nonlinear parabolic problems I,"*SIAM J. Numer. Anal.*, v. 24, 1987, pp. 12-23. MR**874731 (88e:65114)****[6]**K. Eriksson, C. Johnson & V. Thomée, "Time discretization of parabolic problems by the discontinuous Galerkin method,"*RAIRO MAN*, v. 19, 1985, pp. 611-643. MR**826227 (87e:65073)****[7]**R. S. Falk & G. R. Richter, "Analysis of a continuous finite element method for hyperbolic equations,"*SIAM J. Numer. Anal.*, v. 24, 1987, pp. 257-278. MR**881364 (88d:65133)****[8]**B. L. Hulme, "Discrete Galerkin and related one-step methods for ordinary differential equations,"*Math. Comp.*, v. 26, 1972, pp. 881-891. MR**0315899 (47:4448)****[9]**B. L. Hulme, "One-step piecewise polynomial Galerkin methods for initial value problems,"*Math. Comp.*, v. 26, 1972, pp. 415-426. MR**0321301 (47:9834)****[10]**P. Jamet, "Galerkin-type approximations which are discontinuous in time for parabolic problems on a variable domain,"*SIAM J. Numer. Anal.*, v. 15, 1978, pp. 912-928. MR**507554 (80e:65102)****[11]**P. Jamet, "Stability and convergence of a generalized Crank-Nicolson scheme on a variable mesh for the heat equation,"*SIAM J. Numer. Anal.*, v. 17, 1980, pp. 530-539. MR**584728 (82j:65073)****[12]**O. A. Ladyženskaja, V. A. Solonnikov & N. N. Ural'ceva,*Linear and Quasilinear Equations of Parabolic Type*, Transl. Math. Monographs, Vol. 23, Amer. Math. Soc., Providence, R. I., 1967. MR**0241822 (39:3159b)****[13]**P. Lesaint & P. A. Raviart, "Finite element collocation methods for first order systems,"*Math. Comp.*, v. 33, 1979, pp. 891-918. MR**528046 (80d:65118)****[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]**M. Luskin & R. Rannacher, "On the smoothing property of the Galerkin method for parabolic equations,"*SIAM J. Numer. Anal.*, v. 19, 1981, pp. 93-113. MR**646596 (83c:65245)****[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]**V. Thomée, "Negative norm estimates and superconvergence in Galerkin methods for parabolic problems,"*Math. Comp.*, v. 34, 1980, pp. 93-113. MR**551292 (81a:65092)****[20]**M. F. Wheeler, "A priori error estimates for Galerkin approximations to parabolic partial differential equations,"*SIAM J. Numer. Anal.*, v. 10, 1973, pp. 723-759. MR**0351124 (50:3613)****[21]**R. Winther, "A stable finite element method for initial-boundary value problems for first-order hyperbolic systems,"*Math. Comp.*, v. 36, 1981, pp. 65-86. MR**595042 (81m:65181)**

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