Linearization Errors in Discrete Goal-Oriented Error Estimation

TL;DR

This work addresses accurate goal-oriented error estimation for nonlinear functionals in nonlinear PDEs solved by continuous Galerkin FE, where traditional two-level adjoint methods neglect linearization errors. It develops a novel discrete adjoint-based estimator $\\eta_2$ that exactly represents the QoI discretization error between two function spaces by accounting for residual and QoI linearization terms, at the cost of solving an auxiliary nonlinear scalar problem. The authors prove that, for quadratic QoIs, the nonlinear scalar can be solved a priori with $\\theta = 1/2$, enabling efficient initialization and, in general, providing a practical path to exact error representation via $\\boldsymbol{z}^{**}$. Through adjoint verification and diverse numerical experiments (nonlinear Poisson and finite-deformation elasticity), the new estimator demonstrates improved effectivity and more efficient adaptive meshes, particularly for nonlinear QoIs, suggesting substantial gains in accuracy with potentially fewer degrees of freedom. The approach offers a robust framework for adjoint-based adaptivity in nonlinear settings and opens avenues for reducing computational cost via recovery techniques and space-choice studies.

Abstract

This paper is concerned with goal-oriented a posteriori error estimation for nonlinear functionals in the context of nonlinear variational problems solved with continuous Galerkin finite element discretizations. A two-level, or discrete, adjoint-based approach for error estimation is considered. The traditional method to derive an error estimate in this context requires linearizing both the nonlinear variational form and the nonlinear functional of interest which introduces linearization errors into the error estimate. In this paper, we investigate these linearization errors. In particular, we develop a novel discrete goal-oriented error estimate that accounts for traditionally neglected nonlinear terms at the expense of greater computational cost. We demonstrate how this error estimate can be used to drive mesh adaptivity. We show that accounting for linearization errors in the error estimate can improve its effectivity for several nonlinear model problems and quantities of interest. We also demonstrate that an adaptive strategy based on the newly proposed estimate can lead to more accurate approximations of the nonlinear functional with fewer degrees of freedom when compared to uniform refinement and traditional adjoint-based approaches.

Figures (19)

• Figure 1: The domain $\Omega$, its initial mesh, and the subdomain $\Omega_s$ (red outline).
• Figure 2: The manufactured solution (left), the $x$-component of the manufactured solution gradient (center), and the $y$-component of the manufactured solution gradient (right) with $20$ linearly-spaced contour lines.
• Figure 3: The adjoint solution $z$ corresponding to the model problem with a manufactured solution when $\alpha=10^{-2}$ for the nonlinear QoIs $\mathcal{J}_2(u)$ (left), $\mathcal{J}_3(u)$ (center), and the $\mathcal{J}_4(u)$ (right) with 20 linearly spaced contour curves.
• Figure 4: The function $f(\theta)$ plotted at $10$ evenly spaced points in $[0,1]$ for the QoI $\mathcal{J}_2(u)$ for the manufactured solution when $\alpha=10^{-2}$.
• Figure 5: Asymptotic behavior of the estimates $\eta_1$ and $\eta_2$ for the QoI $\mathcal{J}_2(u)$ for the manufactured solution when $\alpha=10^{-2}$ on a sequence of uniformly refined meshes.

Theorems & Definitions (2)

• Proposition 1
• proof