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 \[ {u_t} + Au = f\quad {\text {in}}\quad \Omega \times [0,T],\] where A is a second order elliptic differential operator. Let ${S_h}$; h small denote a family of finite element subspaces of ${H^1}(\Omega )$ which permits approximation of a smooth function to order $O({h^r})$. Let ${\Omega _0} \subset \Omega$ and assume that ${u_h}:[0,T] \to {S_h}$ is an approximate solution which satisfies the semidiscrete interior equation \[ ({u_{h,t}},\chi ) + A({u_h},\chi ) = (f,\chi )\quad \forall \chi \in S_h^0({\Omega _0}) = \{ \chi \in {S_h},{\text {supp}}\chi \subset {\Omega _0}\} ,\] where $A( \cdot , \cdot )$ denotes the bilinear form on ${H^1}(\Omega )$ associated with A. It is shown that if the finite element spaces are based on uniform partitions in a specific sense in ${\Omega _0}$, then difference quotients of ${u_h}$ may be used to approximate derivatives of u in the interior of ${\Omega _0}$ to order $O({h^r})$ provided certain weak global error estimates for ${u_h} - u$ to this order are available. This generalizes results proved for elliptic problems by Nitsche and Schatz [9) and Bramble, Nitsche and Schatz [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, DOI https://doi.org/10.1090/S0025-5718-1975-0398120-7
- 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 448926, DOI https://doi.org/10.1137/0714015
- Jim Douglas Jr. and Todd Dupont, Galerkin methods for parabolic equations, SIAM J. Numer. Anal. 7 (1970), 575–626. MR 277126, DOI https://doi.org/10.1137/0707048
- Jim Douglas Jr. and Todd Dupont, Galerkin methods for parabolic equations with nonlinear boundary conditions, Numer. Math. 20 (1972/73), 213–237. MR 319379, DOI https://doi.org/10.1007/BF01436565
- Todd Dupont, Some $L^{2}$ 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
- G. Fix and N. Nassif, On finite element approximations to time-dependent problems, Numer. Math. 19 (1972), 127–135. MR 311122, DOI https://doi.org/10.1007/BF01402523 H.-P. HELFRICH, Lokale Konvergenz des Galerkinverfahrens bei Gleichungen vom parabolischen Typ in Hilberträumen, Habilitationsschrift, Freiburg, 1975. J. L. LIONS, & E. MAGENES, Non-Homogeneous Boundary Value Problems and Applications, vol. 1, Springer-Verlag, Berlin and New York, 1972.
- Joachim A. Nitsche and Alfred H. Schatz, Interior estimates for Ritz-Galerkin methods, Math. Comp. 28 (1974), 937–958. MR 373325, DOI https://doi.org/10.1090/S0025-5718-1974-0373325-9
- 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
- Mary Fanett Wheeler, A priori $L_{2}$ error estimates for Galerkin approximations to parabolic partial differential equations, SIAM J. Numer. Anal. 10 (1973), 723–759. MR 351124, DOI https://doi.org/10.1137/0710062
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© Copyright 1979
American Mathematical Society