Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

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



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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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].

References [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65N30, 65M15

Retrieve articles in all journals with MSC: 65N30, 65M15

Additional Information

Article copyright: © Copyright 1979 American Mathematical Society