A Note on the Reliability of Goal-Oriented Error Estimates for Galerkin Finite Element Methods with Nonlinear Functionals

TL;DR

This paper investigates the reliability of goal-oriented error estimates in Galerkin finite element methods for nonlinear functionals, showing that such estimates can be unreliable even with exact adjoint solutions.

Contribution

It demonstrates that common error estimates may fail to reliably bound the true error for certain nonlinear functionals, highlighting limitations in existing error estimation techniques.

Findings

Some nonlinear functionals lead to unreliable error estimates.

Reliability of error estimates cannot be guaranteed universally.

Examples show failure of standard estimates even with exact adjoint solutions.

Abstract

We consider estimating the discretization error in a nonlinear functional $J(u)$ in the setting of an abstract variational problem: find $u \in \mathcal{V}$ such that $B(u,\varphi) = L(\varphi) \; \forall \varphi \in \mathcal{V}$, as approximated by a Galerkin finite element method. Here, $\mathcal{V}$ is a Hilbert space, $B(\cdot,\cdot)$ is a bilinear form, and $L(\cdot)$ is a linear functional. We consider well-known error estimates $\eta$ of the form $J(u) - J(u_h) \approx \eta = L(z) - B(u_h, z)$, where $u_h$ denotes a finite element approximation to $u$, and $z$ denotes the solution to an auxiliary adjoint variational problem. We show that there exist nonlinear functionals for which error estimates of this form are not reliable, even in the presence of an exact adjoint solution solution $z$. An estimate $\eta$ is said to be reliable if there exists a constant $C \in \mathbb{R}_{>0}$ independent of $u_h$ such that $|J(u) - J(u_h)| \leq C|\eta|$. We present several example pairs of bilinear forms and nonlinear functionals where reliability of $\eta$ is not achieved.