A quasiprojection analysis of Galerkin methods for parabolic and hyperbolic equations
Authors:
Jim Douglas, Todd Dupont and Mary F. Wheeler
Journal:
Math. Comp. 32 (1978), 345362
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
(50 #15360)
 [2]
Jim
Douglas Jr. and Todd
Dupont, Collocation methods for parabolic equations in a single
space variable, Lecture Notes in Mathematics, Vol. 385,
SpringerVerlag, BerlinNew York, 1974. Based on
𝐶¹piecewisepolynomial spaces. MR 0483559
(58 #3551)
 [3]
J. DOUGLAS, JR., T. DUPONT & M. F. WHEELER, A QuasiProjection 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. R2, 47–59
(English, with Loose French summary). MR 0359357
(50 #11811)
 [5]
Jim
Douglas Jr., Todd
Dupont, and Mary
Fanett Wheeler, Some superconvergence results for an
𝐻¹Galerkin 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
(56 #10056)
 [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
(53 #7067)
 [7]
H.
H. Rachford Jr., Twolevel discretetime Galerkin approximations
for second order nonlinear parabolic partial differential equations,
SIAM J. Numer. Anal. 10 (1973), 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]
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
(50 #3613)
 [10]
Mary
Fanett Wheeler, 𝐿_{∞} estimates of optimal orders for
Galerkin methods for onedimensional second order parabolic and hyperbolic
equations, SIAM J. Numer. Anal. 10 (1973),
908–913. MR 0343658
(49 #8398)
 [11]
Mary
Fanett Wheeler, A Galerkin procedure for estimating the flux for
twopoint boundary value problems, SIAM J. Numer. Anal.
11 (1974), 764–768. MR 0383764
(52 #4644)
 [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
(54 #1670)
 [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. 99109. 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, SpringerVerlag, Berlin and New York, 1974. MR 0483559 (58:3551)
 [3]
 J. DOUGLAS, JR., T. DUPONT & M. F. WHEELER, A QuasiProjection 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. 4759. 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, SpringerVerlag, 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., "Twolevel discretetime Galerkin approximations for second order nonlinear parabolic partial differential equations," SIAM J. Numer. Anal., v. 10, 1973, pp. 10101026. 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. 723759. MR 0351124 (50:3613)
 [10]
 M. F. WHEELER, " estimates of optimal order for Galerkin methods for onedimensional second order parabolic and hyperbolic equations," SIAM J. Numer. Anal., v. 10, 1973, pp. 908913. 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. 764768. 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. 803817. MR 0413556 (54:1670)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718197804950122
PII:
S 00255718(1978)04950122
Article copyright:
© Copyright 1978
American Mathematical Society
