Two-side a posteriori error estimates for the DWR method

TL;DR

This paper develops two-sided a posteriori error estimates for the DWR method, applicable to nonlinear PDEs and multiple goal functionals, with theoretical analysis and numerical validation.

Contribution

It introduces new two-sided error bounds for the DWR method, including for nonlinear problems and multiple functionals, with detailed analysis of the remainder term.

Findings

Derived lower bounds for error estimators confirming efficiency.

Validated theoretical results with numerical experiments.

Balanced discretization and nonlinear iteration errors in algorithms.

Abstract

In this work, we derive two-sided a posteriori error estimates for the dual-weighted residual (DWR) method. We consider both single and multiple goal functionals. Using a saturation assumption, we derive lower bounds yielding the efficiency of the error estimator. These results hold true for both nonlinear partial differential equations and nonlinear functionals of interest. Furthermore, the DWR method employed in this work accounts for balancing the discretization error with the nonlinear iteration error. We also perform careful studies of the remainder term that is usually neglected. Based on these theoretical investigations, several algorithms are designed. Our theoretical findings and algorithmic developments are substantiated with some numerical tests.