### A
posteriori error estimates including algebraic error: Computable upper bounds
and stopping criteria for iterative silvers

**Auteur(s): **

**Le document est une prépublication**
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**Résumé:**

**Abstract :**
We consider the finite volume and the lowest-order mixed finite
element discretizations of a second-order elliptic pure diffusion model problem.
The first goal of this paper is to derive guaranteed and fully computable a
posteriori error estimates which take into account an inexact solution of the
associated linear algebraic system. We show that the algebraic error can be
simply bounded using the algebraic residual vector. Much better results are,
however, obtained using the complementary energy of an equilibrated
Raviart-Thomas-Nédélec discrete vector field whose divergence is
given by a proper weighting of the residual vector. The second goal of this
paper is to construct efficient stopping criteria for iterative solvers such as
the conjugate gradients, GMRES, or Bi-CGStab. We claim that the discretization
error, implied by the given numerical method, and the algebraic one should be in
balance, or, more precisely, that it is enough to solve the linear algebraic
system to the accuracy which guarantees that the algebraic part of the error
does not contribute significantly to the whole error. Our estimates allow a
reliable and cheap comparison of the discretization and algebraic errors. One
can thus use them to stop the iterative algebraic solver at the desired accuracy
level, without performing an excessive number of unnecessary additional
iterations. Under the assumption of the relative balance between the two errors,
we also prove the efficiency of our a posteriori estimates, i.e., we show that
they also represent a lower bound, up to a generic constant, for the overall
energy error. A local version of this result is also stated. Several numerical
experiments illustrate the theoretical results.

**Mots Clés:** *Second-order elliptic partial differential equation; Finite volume method; Mixed finite element method; A posteriori error estimates; Iterative methods for linear algebraic systems; Stopping criteria. *

**Date:** 2008-10-06