Global and Interior Pointwise best Approximation Results for the Gradient of Galerkin Solutions for Parabolic Problems

Abstract

In this paper we establish a best approximation property of fully discrete Galerkin solutions of second-order parabolic problems on convex polygonal and polyhedral domains in the $L^\infty(I;W^{1,\infty}(\Omega))$ norm. The discretization method consists of continuous Lagrange finite elements in space and discontinuous Galerkin methods of arbitrary order in time. The method of the proof differs from the established fully discrete error estimate techniques and uses only elliptic results and discrete maximal parabolic regularity for discontinuous Galerkin methods established by the authors [D. Leykekhman and B. Vexler, Numer. Math., 135 (2017), pp. 923--952]. 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 more local dependence of the error at a point.

Keywords

  1. optimal control
  2. pointwise control
  3. parabolic problems
  4. finite elements
  5. discontinuous Galerkin
  6. error estimates
  7. pointwise error estimates

MSC codes

  1. 65N30

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Published In

cover image SIAM Journal on Numerical Analysis
SIAM Journal on Numerical Analysis
Pages: 2025 - 2049
ISSN (online): 1095-7170

History

Submitted: 16 June 2016
Accepted: 28 March 2017
Published online: 22 August 2017

Keywords

  1. optimal control
  2. pointwise control
  3. parabolic problems
  4. finite elements
  5. discontinuous Galerkin
  6. error estimates
  7. pointwise error estimates

MSC codes

  1. 65N30

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