A posteriori error estimates for elliptic problems in two and three space dimensions

Folkmar A. Bornemann, Bodo Erdmann, Ralf Kornhuber

Research output: Contribution to journalArticlepeer-review

115 Scopus citations

Abstract

Let u ∈ H be the exact solution of a given selfadjoint elliptic boundary value problem, which is approximated by some ũ ∈ S, S being a suitable finite-element space. Efficient and reliable a posteriori estimates of the error ||u - ũ||, measuring the (local) quality of ũ, play a crucial role in termination criteria and in the adaptive refinement of the underlying mesh. A well-known class of error estimates can be derived systematically by localizing the discretized defect problem by using domain decomposition techniques. In this paper, we provide a guideline for the theoretical analysis of such error estimates. We further clarify the relation to other concepts. Our analysis leads to new error estimates, which are specially suited to three space dimensions. The theoretical results are illustrated by numerical computations.

Original languageEnglish
Pages (from-to)1188-1204
Number of pages17
JournalSIAM Journal on Numerical Analysis
Volume33
Issue number3
DOIs
StatePublished - Jun 1996
Externally publishedYes

Keywords

  • A posteriori error estimates
  • Adaptive finite-element methods

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