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Mathematics of Computation

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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

$\displaystyle {u_t} + Au = f\quad {\text{in}}\quad \Omega \times [0,T],$

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

$\displaystyle ({u_{h,t}},\chi ) + A({u_h},\chi ) = (f,\chi )\quad \forall \chi ... ..._h^0({\Omega _0}) = \{ \chi \in {S_h},{\text{supp}}\chi \subset {\Omega _0}\} ,$

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].

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Article copyright: © Copyright 1979 American Mathematical Society