A quasi-projection analysis of Galerkin methods for parabolic and hyperbolic equations

Authors:
Jim Douglas, Todd Dupont and Mary F. Wheeler

Journal:
Math. Comp. **32** (1978), 345-362

MSC:
Primary 65N30; Secondary 65M15

DOI:
https://doi.org/10.1090/S0025-5718-1978-0495012-2

MathSciNet review:
0495012

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Abstract: Superconvergence phenomena are demonstrated for Galerkin approximations of solutions of second order parabolic and hyperbolic problems in a single space variable. An asymptotic expansion of the Galerkin solution is used to derive these results and, in addition, to show optimal order error estimates in Sobolev spaces of negative index in multiple dimensions.

**[1]**J. DOUGLAS, JR. & T. DUPONT, "Galerkin approximations for the two point boundary problem using continuous, piecewise polynomial spaces,"*Numer. Math.*, v. 22, 1974, pp. 99-109. MR**0362922 (50:15360)****[2]**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)****[3]**J. DOUGLAS, JR., T. DUPONT & M. F. WHEELER,*A Quasi-Projection Approximation Method Applied to Galerkin Procedures for Parabolic and Hyperbolic Equations*, Math. Res. Center Rep. #1465, 1974.**[4]**J. DOUGLAS, JR., T. DUPONT & M. F. WHEELER, "A Galerkin procedure for approximating the flux on the boundary for elliptic and parabolic boundary value problems,"*Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge*, v. 8, 1974, pp. 47-59. MR**0359357 (50:11811)****[5]**J. DOUGLAS, JR., T. DUPONT & M. F. WHEELER,*Some Superconvergence Results for an*-*Galerkin Procedure for the Heat Equation*, Lecture Notes in Comput. Sci., vol. 10, Springer-Verlag, Berlin and New York, 1974. MR**0451774 (56:10056)****[6]**T. DUPONT, "Some estimates for parabolic Galerkin methods,"*The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations*(A. K. Aziz, Ed.), Academic Press, New York, 1972. MR**0403255 (53:7067)****[7]**H. H. RACHFORD, JR., "Two-level discrete-time Galerkin approximations for second order nonlinear parabolic partial differential equations,"*SIAM J. Numer. Anal.*, v. 10, 1973, pp. 1010-1026. MR**0339519 (49:4277)****[8]**J. A. WHEELER,*Simulation of Heat Transfer From a Warm Pipeline Buried in Permafrost*, presented to the 74th National Meeting of the American Institute of Chemical Engineers, New Orleans, March, 1973.**[9]**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)****[10]**M. F. WHEELER, " estimates of optimal order for Galerkin methods for one-dimensional second order parabolic and hyperbolic equations,"*SIAM J. Numer. Anal.*, v. 10, 1973, pp. 908-913. MR**0343658 (49:8398)****[11]**M. F. WHEELER, "A Galerkin procedure for estimating the flux for two point boundary problems,"*SIAM J. Numer. Anal.*, v. 11, 1974, pp. 764-768. MR**0383764 (52:4644)****[12]**M. F. WHEELER, "An Galerkin method for parabolic equations in a single space variable,"*SIAM J. Numer. Anal.*, v. 12, 1975, pp. 803-817. MR**0413556 (54:1670)**

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DOI:
https://doi.org/10.1090/S0025-5718-1978-0495012-2

Article copyright:
© Copyright 1978
American Mathematical Society