Exact a posteriori error control for variational problems via convex duality and explicit flux reconstruction

TL;DR

This work develops an exact, computable a posteriori error framework for convex variational problems by leveraging continuous and discrete convex duality and a flux reconstruction mechanism. Central to the approach is a generalized Prager–Synge identity that equates the inaccessible primal–dual error with an accessible primal–dual gap estimator, providing a reliable, localizable error indicator. The authors formulate discrete primal and dual problems based on nonconforming CR and RT elements and derive discrete reconstruction (Marini) formulas that enable simultaneous primal and dual computations with minimal cost. They demonstrate equivalence to residual-type estimators in many model problems and validate the methodology through extensive numerical experiments across nonlinear Dirichlet, obstacle, Signorini, ROF, jumping coefficients, elasto-plastic torsion, and Stokes problems, including anisotropic and adaptive mesh refinement. The framework offers a broadly applicable and robust route to error control in non-smooth convex minimization, with practical procedures for admissible flux reconstruction and node-averaging post-processing.

Abstract

A posteriori error estimates are an important tool to bound discretization errors in terms of computable quantities avoiding regularity conditions that are often difficult to establish. For non-linear and non-differentiable problems, problems involving jumping coefficients, and finite element methods using anisotropic triangulations, such estimates often involve large factors, leading to sub-optimal error estimates. By making use of convex duality arguments, exact and explicit error representations are derived that avoid such effects.

Figures (21)

• Figure 1: LEFT: primal-dual gap estimator ${\eta^2_{\textup{gap}}}(\overline{u}_k^{cr},z_k^{rt})$ and alternative total error $\tilde{\rho}^2_{\textup{tot}}(\overline{u}_k^{cr},z_k^{rt})$; RIGHT: primal energy $I(\overline{u}_k^{cr})$ and dual energy $D(z_k^{rt})$; each for $p^-\in\{1.5,2\}$ and $k=0,\dots,20$, when using adaptive mesh refinement (i.e., $\theta=\frac{1}{2}$ in Algorithm \ref{['alg:afem']}), and for $k=0,\dots, 4$, when uniform mesh refinement (i.e., $\theta\space=\space1$ in Algorithm \ref{['alg:afem']}), in the $p(\cdot)$-Dirichlet~problem.
• Figure 2: Initial triangulation $\mathcal{T}_0$ and adaptively refined triangulations $\mathcal{T}_k$, ${k\space\in\space \{10,20\}}$, generated by Algorithm \ref{['alg:afem']} for $\theta=\frac{1}{2}$, each in the case $p^-=2$ in the $p(\cdot)$-Dirichlet problem.
• Figure 3: LEFT: discrete primal solution $u_{10}^{cr}\in \mathcal{S}^{1,cr}_0(\mathcal{T}_{10})$; MIDDLE: node-averaged discrete primal solution $\overline{u}_{10}^{cr}\in \mathcal{S}^1_0(\mathcal{T}_{10})$; RIGHT: (local) $L^2$-projection (onto $(\mathcal{L}^0(\mathcal{T}_{10}))^2$) of discrete dual solution $z_{10}^{rt}\in \mathcal{R}T^0(\mathcal{T}_{10})$, each in the the case $p^-=2$ in the $p(\cdot)$-Dirichlet problem.
• Figure 4: LEFT: nodal interpolant of the obstacle $\chi\in W^{1,2}_0(\Omega)$; RIGHT: (local) $L^2$-projection (onto $\mathcal{L}^0(\mathcal{T}_h)$) $\chi_{15}\coloneqq \Pi_{h_{15}}\chi \in \mathcal{L}^0(\mathcal{T}_{15})$ of the obstacle $\chi\in W^{1,2}_0(\Omega)$.
• Figure 5: Adaptively refined triangulations $\mathcal{T}_k$, $k\in\{0,5,10,15,20,25\}$, with discrete contact zones $\mathcal{C}_k^{cr}$, $k\in\{0,5,10,15,20,25\}$, shown in white in the obstacle problem.

Theorems & Definitions (28)

• Definition 2.1: Brègman distance and symmetric Brègman distance
• Definition 2.2: Optimal convexity measure at a minimizer
• Remark 2.3
• Proposition 3.1: Strong duality and convex duality relations
• proof
• Remark 3.2: Equivalent convex optimality relations
• Theorem 3.3: Generalized Prager--Synge identity
• proof
• Remark 3.4
• Proposition 3.5: Strong duality and convex duality relations