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Variational problems with gradient constraints: $\textit{A priori}$ and $\textit{a posteriori}$ error identities

Harbir Antil, Sören Bartels, Alex Kaltenbach, Rohit Khandelwal

TL;DR

An $\textit{a priori}$ error identity is derived that applies to the approximation of the primal formulation using the Crouzeix-Raviart element and to the approximation of the dual formulation using the Raviart-Thomas element and leads to error decay rates that are optimal with respect to the regularity of a dual solution.

Abstract

In this paper, on the basis of a (Fenchel) duality theory on the continuous level, we derive an $\textit{a posteriori}$ error identity for arbitrary conforming approximations of a primal formulation and a dual formulation of variational problems involving gradient constraints. In addition, on the basis of a (Fenchel) duality theory on the discrete level, we derive an $\textit{a priori}$ error identity that applies to the approximation of the primal formulation using the Crouzeix-Raviart element and to the approximation of the dual formulation using the Raviart-Thomas element, and leads to error decay rates that are optimal with respect to the regularity of a dual solution.

Variational problems with gradient constraints: $\textit{A priori}$ and $\textit{a posteriori}$ error identities

TL;DR

An error identity is derived that applies to the approximation of the primal formulation using the Crouzeix-Raviart element and to the approximation of the dual formulation using the Raviart-Thomas element and leads to error decay rates that are optimal with respect to the regularity of a dual solution.

Abstract

In this paper, on the basis of a (Fenchel) duality theory on the continuous level, we derive an error identity for arbitrary conforming approximations of a primal formulation and a dual formulation of variational problems involving gradient constraints. In addition, on the basis of a (Fenchel) duality theory on the discrete level, we derive an error identity that applies to the approximation of the primal formulation using the Crouzeix-Raviart element and to the approximation of the dual formulation using the Raviart-Thomas element, and leads to error decay rates that are optimal with respect to the regularity of a dual solution.

Paper Structure

This paper contains 21 sections, 12 theorems, 141 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Main contributions
  3. Related contributions
  4. Preliminaries
  5. Classical function spaces
  6. Integration-by-parts formula
  7. Bounded and finitely additive vector measures
  8. Triangulations and standard finite element spaces
  9. Crouzeix--Raviart element
  10. Raviart--Thomas element
  11. Discrete integration-by-parts formula
  12. Gradient constrained variational problem
  13. A posteriori error analysis
  14. Discrete gradient constrained variational problem
  15. A priori error analysis
  16. ...and 6 more sections

Key Result

Theorem 3.1

The following statements apply:

Figures (3)

  • Figure 1: Logarithmic plots of the experimental convergence rates of the error quantities \ref{['eq:error_quantities']}. For forcing terms $C\in \{2.5,5.0,7.5,10.0\}$, we report the expected quadratic error decay rate, i.e., $\texttt{EOC}_i(e_i^{\textup{tot}})\approx\texttt{EOC}_i(e_i^{\textup{gap}})\approx 2$, ${i=1,\ldots, 6}$. In addition, we report that the discrete primal-dual error (cf. \ref{['eq:discrete_primal_dual_error']}) approximatively coincides with the discrete primal-dual gap estimator (cf. \ref{['eq:discrete_primal_dual_gap_estimator']}).
  • Figure 2: left: plot of $\vert \nabla_{h_2}u_{h_2}^{cr}\vert\in \mathcal{L}^0(\mathcal{T}_{h_2})$, where red dots mark $T\in \mathcal{T}_{h_2}$ with $\vert \nabla_{h_2}u_{h_2}^{cr}\vert=1$ in $T$; right: plot of $\vert \Pi_{h_2} z_{h_2}^{rt}\vert\space\in\space \mathcal{L}^0(\mathcal{T}_{h_2})$, where red dots mark $T\space\in \space\mathcal{T}_{h_2}$ with ${\vert \Pi_{h_2} z_{h_2}^{rt}\vert\space<\space1}$ in $T$. In summary, We report that $\{\vert \nabla_{h_2} u_{h_2}^{cr}\vert=1\}=\{\vert \Pi_{h_2} z_{h_2}^{rt}\vert\ge 1\}$ and $\{\vert \nabla_{h_2} u_{h_2}^{cr}\vert<1\}=\{\vert \Pi_{h_2} z_{h_2}^{rt}\vert< 1\}$ as predicted by the discrete convex optimality relation \ref{['eq:discrete_optimality']}.
  • Figure 3: Logarithmic plots of the experimental convergence rates of the error quantities \ref{['eq:error_quantities']}. For $C=10.0$ and $\Pi_{h_i}^{w1}\in \{\Pi_{h_i}^{av,1},\Pi_{h_i}^{av,2},\Pi_{h_i}^{sz,1},\Pi_{h_i}^{sz,2},\Pi_{h_i}^{L^2,1},\Pi_{h_i}^{L^2,2}\}$, we report a reduced linear error decay rate, i.e., $\texttt{EOC}_i(e_i^{\textup{tot}})\approx\texttt{EOC}_i(e_i^{\textup{gap}})\approx 1$, ${i=1,\ldots, 6}$. In addition, we report that the primal-dual error (cf. \ref{['def:primal_dual_total_error']}) approximatively coincides with the primal-dual gap estimator (cf. \ref{['eq:primal-dual.1']}).

Theorems & Definitions (32)

  • Theorem 3.1: strong duality and convex optimality relations
  • Remark 3.2
  • proof : Proof (of Theorem \ref{['thm:duality']}).
  • Lemma 4.1: decomposition of the primal-dual gap estimator
  • Remark 4.2: interpretation of the components of the primal-dual gap estimator
  • proof : Proof (of Lemma \ref{['lem:primal_dual_gap_estimator']}).
  • Lemma 4.3: representations of the optimal strong convexity measures
  • Remark 4.4
  • proof : Proof (of Lemma \ref{['lem:strong_convexity_measures']})
  • Theorem 4.5: primal-dual gap identity
  • ...and 22 more