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

TL;DR

This paper introduces guaranteed lower bounds for error estimation in finite element and domain decomposition methods, effectively separating algebraic and discretization errors to improve solver efficiency and goal-oriented error control.

Contribution

It presents a novel approach to compute fully parallel lower bounds that distinguish between algebraic and discretization errors, enhancing error estimation and solver steering in domain decomposition methods.

Findings

Lower bounds are independent of substructuring.

Separation of algebraic and discretization errors improves error control.

Adaptive remeshing and recycling enhance solver precision.

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.