A posteriori estimates and stopping criteria for iterative linearizations and linear solvers

• ID: 2621, RIV: 10107093
• ISSN: not specified, ISBN: 978-80-7372-724-6
• source: International Conference Presentation of Mathematics '10
• keywords: nonlinear problem; finite volume method; linearization; iterative methods; a posteriori error estimate; stopping criteria
• authors: Linda El Alaoui, Alexandre Ern, Pavel Jiránek, Zdeněk Strakoš, Martin Vohralík
• authors from KNM: Strakoš Zdeněk

Abstract

We present the a posteriori error estimates which enable to take into account the linearization error in approximation of nonlinear problems and the algebraic error in the solution of linear systems associated to the given numerical discretization. Our estimates allow to distinguish, estimate separately, and compare these different error sources. Consequently, the iterative (Newton, quasi-Newton) linearization or iterative solution of linear algebraic systems can be stopped whenever the individual errors drop to the level at which they do not affect significantly the overall error. This can lead to important computational savings, as performing an excessive number of unnecessary linearization/linear solver iterations can be avoided. Moreover, due to their local efficiency, our estimators also allow to accurately predict the error spatial distribution and thus they are suitable for local adaptive mesh refinement. Finally, they give a fully computable upper bound on the overall error. This allows to devise an adaptive strategy enabling to achieve a user-given precision at minimal cost. Numerical experiments illustrate the theoretical developments.