Error estimation for FETI(DP) and BDD(C)
Résumé
We consider linear elasticity problems approximated by the Finite Element (FE) method with the resulting linear systems being solved by non-overlapping domain decomposition methods like FETI(DP) or BDD(C). This presentation deals with the computation of guaranteed upper and lower bounds of the error during the iterations of the solver.
The bounds we consider are based on the error in constitutive equation and on the residual equation. These methods imply to recover certain displacement and stress fields with high regularity on the whole structure. We first show that it is possible to intercept, during the iterations of FETI(DP) or BDD(C), the information necessary for the parallel construction of such fields.
Then by a little modification of the classical bounds, we manage to obtain new bounds which separate the algebraic error (due to the use of a DD iterative solver) from the discretization error (due to the FE) . These bounds provides an unbiased criterion to stop the iterations when the solver error is lesser than the discretization error. They can also be used for the estimation of the error on linear quantities of interest.
Assessments on 2D static linear mechanic problems illustrate the relevance of the separation of sources of error and the independence of the bounds with respect to the substructuring. Finally, we steer the iterative solver by an objective of precision on a quantity of interest. The strategy consists in a sequence of solving, adaptive local remeshing and recycling of search directions, in order to reach the desired quality for the quantity of interest with minimal computations.
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