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

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]**James H. Bramble, Joachim A. Nitsche, and Alfred H. Schatz,*Maximum-norm interior estimates for Ritz-Galerkin methods*, Math. Comput.**29**(1975), 677–688. MR**0398120**, 10.1090/S0025-5718-1975-0398120-7**[2]**J. H. Bramble, A. H. Schatz, V. Thomée, and L. B. Wahlbin,*Some convergence estimates for semidiscrete Galerkin type approximations for parabolic equations*, SIAM J. Numer. Anal.**14**(1977), no. 2, 218–241. MR**0448926****[3]**Jim Douglas Jr. and Todd Dupont,*Galerkin methods for parabolic equations*, SIAM J. Numer. Anal.**7**(1970), 575–626. MR**0277126****[4]**Jim Douglas Jr. and Todd Dupont,*Galerkin methods for parabolic equations with nonlinear boundary conditions*, Numer. Math.**20**(1972/73), 213–237. MR**0319379****[5]**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****[6]**G. Fix and N. Nassif,*On finite element approximations to time-dependent problems*, Numer. Math.**19**(1972), 127–135. MR**0311122****[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]**Joachim A. Nitsche and Alfred H. Schatz,*Interior estimates for Ritz-Galerkin methods*, Math. Comp.**28**(1974), 937–958. MR**0373325**, 10.1090/S0025-5718-1974-0373325-9**[10]**Harvey S. Price and Richard S. Varga,*Error bounds for semidiscrete Galerkin approximations of parabolic problems with applications to petroleum reservoir mechanics*, Numerical Solution of Field Problems in Continuum Physics (Proc. Sympos. Appl. Math., Durham, N.C., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 74–94. MR**0266452****[11]**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**

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

Article copyright:
© Copyright 1979
American Mathematical Society