Strict bounding of quantities of interest in computations based on domain decomposition

TL;DR

This paper introduces a method for tightly bounding errors in quantities of interest in finite element and domain decomposition computations, effectively separating discretization and algebraic errors for improved accuracy and efficiency.

Contribution

It proposes new bounds that distinguish discretization and algebraic errors, allowing for relaxed interface conformity and local refinement in domain decomposition methods.

Findings

Bounds effectively separate discretization and algebraic errors

Relaxed interface conformity enables local enrichment

Validated on 2D static linear mechanics problems

Abstract

This paper deals with bounding the error on the estimation of quantities of interest obtained by finite element and domain decomposition methods. The proposed bounds are written in order to separate the two errors involved in the resolution of reference and adjoint problems : on the one hand the discretization error due to the finite element method and on the other hand the algebraic error due to the use of the iterative solver. Beside practical considerations on the parallel computation of the bounds, it is shown that the interface conformity can be slightly relaxed so that local enrichment or refinement are possible in the subdomains bearing singularities or quantities of interest which simplifies the improvement of the estimation. Academic assessments are given on 2D static linear mechanic problems.