Fully Reliable Localized Error Control in the FEM

  • Carstensen C
  • Funken S
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If the first task in numerical analysis is the calculation of an approximate

the second is to provide a guaranteed error bound and is often of
equal importance. The standard

approaches in the a posteriori error analysis of finite element methods
suppose that the exact solution

has a certain regularity or the numerical scheme enjoys some saturation
property. For coarse meshes

those asymptotic arguments are difficult to recast into rigorous error
bounds. The aim of this paper

is to provide reliable computable error bounds which are efficient
and complete in the sense that

constants are estimated as well. The main argument is a localization
via a partition of unity which

leads to problems on small domains. Two fully reliable estimates are
established: The sharper

one solves an analytical interface problem with residuals following
Babuˇka and Rheinboldt [SIAM


J. Numer. Anal., 15 (1978), pp. 736–754]. The second estimate is a
modification of the standard

residual-based a posteriori estimate with explicit constants from
local analytical eigenvalue problems.

For some class of triangulations we show that the efficiency constant
is smaller than 2.5. According

to our numerical experience, the overestimation of our computable
estimates proved to be reasonably

small, with an overestimation by a factor between 2.5 and 4 only.

Author-supplied keywords

  • 35j70
  • 65n30
  • 73c60
  • a posteriori error estimates
  • adaptive
  • algorithms
  • ams subject classifications
  • computable error bounds
  • error control
  • pii
  • reliability
  • s1064827597327486

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  • Carsten Carstensen

  • Stefan a. Funken

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