Worst-case multi-objective error estimation and adaptivity

TL;DR

This paper develops a novel methodology for worst-case multi-objective error estimation in finite-element methods, enabling adaptive refinement based on a comprehensive error measure that considers multiple objectives simultaneously.

Contribution

It introduces a general approach for a-posteriori error estimation for multiple objectives, extending classical goal-oriented methods to a worst-case framework with adaptive refinement capabilities.

Findings

Effective in guiding adaptive finite-element refinements

Handles multiple objectives simultaneously

Demonstrates improved error control in numerical experiments

Abstract

This paper introduces a new computational methodology for determining a-posteriori multi-objective error estimates for finite-element approximations, and for constructing corresponding (quasi-)optimal adaptive refinements of finite-element spaces. As opposed to the classical goal-oriented approaches, which consider only a single objective functional, the presented methodology applies to general closed convex subsets of the dual space and constructs a worst-case error estimate of the finite-element approximation error. This worst-case multi-objective error estimate conforms to a dual-weighted residual, in which the dual solution is associated with an approximate supporting functional of the objective set at the approximation error. We regard both standard approximation errors and data-incompatibility errors associated with incompatibility of boundary data with the trace of the finite-element space. Numerical experiments are presented to demonstrate the efficacy of applying the proposed worst-case multi-objective error in adaptive refinement procedures.