Some interior estimates for semidiscrete Galerkin approximations for parabolic equations

Author:
Vidar Thomée

Journal:
Math. Comp. **33** (1979), 37-62

MSC:
Primary 65N30; Secondary 65M15

DOI:
https://doi.org/10.1090/S0025-5718-1979-0514809-4

MathSciNet review:
514809

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Abstract: Consider a solution *u* of the parabolic equation

*A*is a second order elliptic differential operator. Let ;

*h*small denote a family of finite element subspaces of which permits approximation of a smooth function to order . Let and assume that is an approximate solution which satisfies the semidiscrete interior equation

*A*. It is shown that if the finite element spaces are based on uniform partitions in a specific sense in , then difference quotients of may be used to approximate derivatives of

*u*in the interior of to order provided certain weak global error estimates for to this order are available. This generalizes results proved for elliptic problems by Nitsche and Schatz [9) and Bramble, Nitsche and Schatz [1].

**[1]**J. H. BRAMBLE, J. A. NITSCHE & A. H. SCHATZ, "Maximum norm interior estimates for Ritz-Galerkin methods,"*Math. Comp.*, v. 29, 1975, pp. 677-688. MR**0398120 (53:1975)****[2]**J. H. BRAMBLE, A. H. SCHATZ, V. THOMÉE & L. B. WAHLBIN, "Some convergence estimates for Galerkin type approximations for parabolic equations,"*SIAM J. Numer. Anal.*, v. 14, 1977, pp. 218-241. MR**0448926 (56:7231)****[3]**J. DOUGLAS, JR. & T. DUPONT, "Galerkin methods for parabolic equations,"*SIAM J. Numer. Anal.*, v. 7, 1970, pp. 575-626. MR**0277126 (43:2863)****[4]**J. DOUGLAS, JR. & T. DUPONT, "Galerkin methods for parabolic equations with nonlinear boundary conditions,"*Numer. Math.*, v. 20, 1973, pp. 213-237. MR**0319379 (47:7923)****[5]**T. DUPONT, "Some error estimates for parabolic Galerkin methods,"*The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations*(A. K. Aziz, Editor), Academic Press, New York, 1972, pp. 491-501. MR**0403255 (53:7067)****[6]**G. FIX & N. NASSIF, "On finite element approximations to time dependent problems,"*Numer. Math.*, v. 19, 1972, pp. 127-135. MR**0311122 (46:10218)****[7]**H.-P. HELFRICH,*Lokale Konvergenz des Galerkinverfahrens bei Gleichungen vom parabolischen Typ in Hilberträumen*, Habilitationsschrift, Freiburg, 1975.**[8]**J. L. LIONS, & E. MAGENES,*Non-Homogeneous Boundary Value Problems and Applications*, vol. 1, Springer-Verlag, Berlin and New York, 1972.**[9]**J. A. NITSCHE & A. H. SCHATZ, "Interior estimates for Ritz-Galerkin methods,"*Math. Comp.*, v. 28, 1974, pp. 937-958. MR**0373325 (51:9525)****[10]**H. S. PRICE & R. S. VARGA, "Error bounds for semi-discrete Galerkin approximations of parabolic problems with application to petroleum reservoir mechanics,"*Numerical Solution of Field Problems in Continuum Physics*, SIAM-AMS Proc., vol. 11, Amer. Math. Soc., Providence, R. I., 1970, pp. 74-94. MR**0266452 (42:1358)****[11]**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)**

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DOI:
https://doi.org/10.1090/S0025-5718-1979-0514809-4

Article copyright:
© Copyright 1979
American Mathematical Society