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


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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