A posteriori error estimates including algebraic error and stopping criteria for iterative solvers

• ID: 2478, RIV: 10051636
• ISSN: 1064-8275, ISBN: not specified
• source: SIAM Journal of Scientific Computing
• keywords: second-order elliptic partial differential equation; finite volume method; a posteriori error estimates; iterative methods for linear algebraic systems; conjugate gradient method; stopping criter
• authors: Pavel Jiránek, Zdeněk Strakoš, Martin Vohralík
• authors from KNM: Strakoš Zdeněk

Abstract

For the finite volume discretization of a second-order elliptic model problem, we derive 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 bounded by constructing an equilibrated Raviart-Thomas-Nédélec discrete vector field whose divergence is given by a proper weighting of the residual vector. Next, claiming that the discretization error and the algebraic one should be in balance, we construct stopping criteria for iterative algebraic solvers. An attention is paid, in particular, to the conjugate gradient method which minimizes the energy norm of the algebraic error. We also prove the efficiency of our a posteriori estimates. A local version of this result is also stated. This makes our approach suitable for adaptive mesh refinement which also takes into account the algebraic error.