Strict lower bounds with separation of sources of error in non‐overlapping domain decomposition methods

Abstract

This article deals with the computation of guaranteed lower bounds of the error in the framework of finite element (FE) and domain decomposition (DD) methods. In addition to a fully parallel computation, the proposed lower bounds separate the algebraic error (due to the use of a DD iterative solver) from the discretization error (due to the FE), which enables the steering of the iterative solver by the discretization error. These lower bounds are also used to improve the goal-oriented error estimation in a substructured context. Assessments on 2D static linear mechanic problems illustrate the relevance of the separation of sources of error and the lower bounds' independence from the substructuring. We also steer the iterative solver by an objective of precision on a quantity of interest. This strategy consists in a sequence of solvings and takes advantage of adaptive remeshing and recycling of search directions.

Improved recovery of admissible stress in domain decomposition methods — application to heterogeneous structures and new error bounds for FETI‐DP

“…Thus all the results involving separation of sources of error (11) can be used in FETI-DP (see also [30,31] for bounds on quantities of interest and lower bounds).…”

“…where we have used the fact that ıu .s/ c D 0 This means that the quantity jjju N u D jjj 2 is naturally computed at each iteration. Thus, all the results involving separation of sources of error (11) can be used in FETI-DP (see also [30,31] for bounds on quantities of interest and lower bounds).…”

“…The results of previous sections are very useful in order to derive a procedure to obtain admissible fields in the FETI-DP method [7,6] and to prove that the bounds with separated contributions of [29,30,31] also apply to this algorithm.…”

A posteriori error estimates and stopping criteria for space-time domain decomposition for two-phase flow between different rock types

“…In practice, this means that the a posteriori error estimates we develop have to hold true at each iteration of the OSWR algorithm and the linearization, and distinguish the different error components (space, time, linearization, and domain decomposition). For this part of our work, we take up the path initiated in [9,11,26,38,54] for general techniques taking into account inexact algebraic solvers, [45] for (multiscale) mortar techniques, [46,47] for FETI and BDD domain decomposition algorithms combined with conforming finite element discretizations, and most closely [6,7] for respectively steady and unsteady linear problems with similar space and time discretizations. To achieve our goals here, we use a) equilibrated H(div)-conforming flux reconstructions, piecewise constant in time, that are direct extension of [7,25,54] to our model; b) novel subdomain-wise H 1 -conforming saturation reconstructions, continuous and piecewise affine in time, which are targeted to the present setting in that they satisfy the nonlinear and discontinuous interface conditions.…”

A Posteriori Stopping Criteria for Optimized Schwarz Domain Decomposition Algorithms in Mixed Formulations

“…Coupling specifically domain decomposition and a posteriori error estimates has also recently been addressed in [43,44]. Here, the case of the linear elasticity problem approximated by the finite element method in combination with non-overlapping domain decomposition method such as the finite element tearing and interconnecting (FETI, see [20]) or Balancing Neumann-Neumann (BDD, see [35] and [12] for the case of mixed finite elements) have been studied.…”