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

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]**Jim Douglas Jr. and Todd Dupont,*Galerkin approximations for the two point boundary problem using continuous, piecewise polynomial spaces*, Numer. Math.**22**(1974), 99–109. MR**0362922****[2]**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****[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]**Jim Douglas Jr., Todd Dupont, and Mary Fanett 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**8**(1974), no. R-2, 47–59 (English, with Loose French summary). MR**0359357****[5]**Jim Douglas Jr., Todd Dupont, and Mary Fanett Wheeler,*Some superconvergence results for an 𝐻¹-Galerkin procedure for the heat equation*, Computing methods in applied sciences and engineering (Proc. Internat. Sympos., Versailles, 1973) Springer, Berlin, 1974, pp. 288–311. Lecture Notes in Comput. Sci., Vol. 10. MR**0451774****[6]**Todd Dupont,*Some 𝐿² error estimates for parabolic Galerkin methods*, 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. 491–504. MR**0403255****[7]**H. H. Rachford Jr.,*Two-level discrete-time Galerkin approximations for second order nonlinear parabolic partial differential equations*, SIAM J. Numer. Anal.**10**(1973), 1010–1026. MR**0339519****[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]**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****[10]**Mary Fanett Wheeler,*𝐿_{∞} estimates of optimal orders for Galerkin methods for one-dimensional second order parabolic and hyperbolic equations*, SIAM J. Numer. Anal.**10**(1973), 908–913. MR**0343658****[11]**Mary Fanett Wheeler,*A Galerkin procedure for estimating the flux for two-point boundary value problems*, SIAM J. Numer. Anal.**11**(1974), 764–768. MR**0383764****[12]**Mary F. Wheeler,*An 𝐻⁻¹ Galerkin method for parabolic problems in a single space variable*, SIAM J. Numer. Anal.**12**(1975), no. 5, 803–817. MR**0413556**

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

Article copyright:
© Copyright 1978
American Mathematical Society