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,
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© Copyright 1979
American Mathematical Society