Pointwise best approximation results for Galerkin finite element solutions of parabolic problems

Dmitriy Leykekhman, Boris Vexler

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21 Scopus citations

Abstract

In this paper we establish a best approximation property of fully discrete Galerkin finite element solutions of second order parabolic problems on convex polygonal and polyhedral domains in the L∞ norm. The discretization method uses continuous Lagrange finite elements in space and discontinuous Galerkin methods in time of an arbitrary order. The method of proof differs from the established fully discrete error estimate techniques and for the first time allows one to obtain such results in three space dimensions. It uses elliptic results, discrete resolvent estimates in weighted norms, and the discrete maximal parabolic regularity for discontinuous Galerkin methods established by the authors in [Numer. Math., submitted; available online at http://arxiv.org/abs/1505.04808/]. In addition, the proof does not require any relationship between spatial mesh sizes and time steps. We also establish an interior best approximation property that shows a more local behavior of the error at a given point.

Original languageEnglish
Pages (from-to)1365-1384
Number of pages20
JournalSIAM Journal on Numerical Analysis
Volume54
Issue number3
DOIs
StatePublished - 2016

Keywords

  • A priori error estimates
  • Discontinuous Galerkin
  • Finite elements
  • Parabolic problems
  • Pointwise error estimates

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