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 $L^\infty$ 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.

Keywords

  1. parabolic problems
  2. finite elements
  3. discontinuous Galerkin
  4. a priori error estimates
  5. pointwise error estimates

MSC codes

  1. 65M60

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Information & Authors

Information

Published In

cover image SIAM Journal on Numerical Analysis
SIAM Journal on Numerical Analysis
Pages: 1365 - 1384
ISSN (online): 1095-7170

History

Submitted: 6 August 2015
Accepted: 12 February 2016
Published online: 10 May 2016

Keywords

  1. parabolic problems
  2. finite elements
  3. discontinuous Galerkin
  4. a priori error estimates
  5. pointwise error estimates

MSC codes

  1. 65M60

Authors

Affiliations

Funding Information

Division of Mathematical Sciences http://dx.doi.org/10.13039/100000121 : 1522555

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