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, adjoint-weighted error estimates for Galerkin FEM when the quantity of interest is a nonlinear functional $J(u)$. It shows that while a standard estimator $\eta_1$ can be reliable under a saturation assumption, two alternative estimators $\eta_2$ and $\eta_3$ can be completely unreliable for certain nonlinear functionals due to Galerkin orthogonality and discrete orthogonality, even with enriched adjoint spaces. The authors provide theoretical constructions (e.g., $J(u)=G(B(u,u))$ and energy-type functionals) and practical Poisson and elasticity examples, including a numerical Poisson test where $\eta_3$ vanishes while the true error remains nonzero. The work highlights the limitations of traditional adjoint-based error control in nonlinear settings and points to future directions, such as linearizing about the exact solution or the fine-space solution, to better capture linearization errors in multi-goal contexts.

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 $η$ of the form $J(u) - J(u_h) \approx η= L(z) - B(u_h, z)$, 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 $η$ is said to be reliable if there exists a constant $C \in \mathbb{R}_{>0}$ independent of $u_h$ such that $|J(u) - J(u_h)| \leq C|η|$. We present several example pairs of bilinear forms and nonlinear functionals where reliability of $η$ is not achieved.

Figures (1)

• Figure 1: For the numerical demonstration: (left) the domain $\Omega$ and computational mesh with size $h=1/16$, (center) the finite element solution $u_h$, (left) a graphical approximation to element-level contributions to the error $J(u) - J(u_h)$ by plotting $|J(u_h^{\texttt{+}}) - J(u_h)| \bigr|_{\Omega^e}$, where $\Omega^e$ corresponds to an individual element.

Theorems & Definitions (22)

• Remark 1
• Definition 1
• Lemma 1
• Lemma 2
• proof
• Lemma 3
• proof
• Remark 2
• Definition 2
• Lemma 4