# Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

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## Some interior estimates for semidiscrete Galerkin approximations for parabolic equationsHTML articles powered by AMS MathViewer

by Vidar Thomée
Math. Comp. 33 (1979), 37-62 Request permission

## 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 .
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