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A Priori and A Posteriori Error Identities for Vectorial Problems via Convex Duality

P. A. Gazca-Orozco, A. Kaltenbach

TL;DR

This work develops a comprehensive convex-duality framework to derive both a priori and a posteriori error identities for vector-valued PDEs, specifically the incompressible Stokes and Navier–Lamé problems, under inhomogeneous mixed boundary conditions and loads in dual spaces. By employing nonconforming Crouzeix–Raviart discretisations for the primal variable and Raviart–Thomas discretisations for the flux/stress, the authors establish discrete Prager–Synge-type identities and a rigorous quasi-optimality theory with minimal regularity assumptions. For elasticity, they introduce an extended dual framework that drops the symmetry requirement, enabling a practical a posteriori equivalence and a Marini-type stress reconstruction that yields an H(div)-conforming stress with comparable cost to the primal solve. Numerical experiments confirm the theoretical results, showcasing reliable error control, explicit constants, and effective adaptive refinement guided by local gap indicators for both Stokes and Navier–Lamé problems. The methodology advances robust, data-oscillation-tolerant error control for vectorial, non-linear or non-smooth generalisations of these foundational PDE systems.

Abstract

Convex duality has been leveraged in recent years to derive a posteriori error estimates and identities for a wide range of non-linear and non-smooth scalar problems. By employing remarkable compatibility properties of the Crouzeix-Raviart and Raviart-Thomas elements, optimal convergence of non-conforming discretisations and flux reconstruction formulas have also been established. This paper aims to extend these results to the vectorial setting, focusing on the archetypical problems of incompressible Stokes and Navier-Lamé. Moreover, unlike most previous results, we consider inhomogeneous mixed boundary conditions and loads in the topological dual of the energy space. At the discrete level, we derive error identities and estimates that enable to prove quasi-optimal error estimates for a Crouzeix-Raviart discretisation with minimal regularity assumptions and no data oscillation terms.

A Priori and A Posteriori Error Identities for Vectorial Problems via Convex Duality

TL;DR

This work develops a comprehensive convex-duality framework to derive both a priori and a posteriori error identities for vector-valued PDEs, specifically the incompressible Stokes and Navier–Lamé problems, under inhomogeneous mixed boundary conditions and loads in dual spaces. By employing nonconforming Crouzeix–Raviart discretisations for the primal variable and Raviart–Thomas discretisations for the flux/stress, the authors establish discrete Prager–Synge-type identities and a rigorous quasi-optimality theory with minimal regularity assumptions. For elasticity, they introduce an extended dual framework that drops the symmetry requirement, enabling a practical a posteriori equivalence and a Marini-type stress reconstruction that yields an H(div)-conforming stress with comparable cost to the primal solve. Numerical experiments confirm the theoretical results, showcasing reliable error control, explicit constants, and effective adaptive refinement guided by local gap indicators for both Stokes and Navier–Lamé problems. The methodology advances robust, data-oscillation-tolerant error control for vectorial, non-linear or non-smooth generalisations of these foundational PDE systems.

Abstract

Convex duality has been leveraged in recent years to derive a posteriori error estimates and identities for a wide range of non-linear and non-smooth scalar problems. By employing remarkable compatibility properties of the Crouzeix-Raviart and Raviart-Thomas elements, optimal convergence of non-conforming discretisations and flux reconstruction formulas have also been established. This paper aims to extend these results to the vectorial setting, focusing on the archetypical problems of incompressible Stokes and Navier-Lamé. Moreover, unlike most previous results, we consider inhomogeneous mixed boundary conditions and loads in the topological dual of the energy space. At the discrete level, we derive error identities and estimates that enable to prove quasi-optimal error estimates for a Crouzeix-Raviart discretisation with minimal regularity assumptions and no data oscillation terms.
Paper Structure (28 sections, 17 theorems, 204 equations, 5 figures, 1 table, 1 algorithm)

This paper contains 28 sections, 17 theorems, 204 equations, 5 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. The incompressible Stokes problem
  3. The Navier--Lamé problem
  4. Preliminaries
  5. Triangulations
  6. The Crouzeix--Raviart element
  7. The Raviart--Thomas element
  8. Discrete integration-by-parts formula
  9. The incompressible Stokes problem
  10. The primal problem
  11. The dual problem
  12. The discrete primal problem
  13. The discrete dual problem
  14. A priori error analysis
  15. A posteriori error analysis
  16. ...and 13 more sections

Key Result

lemma thmcounterlemma

For every load $\boldsymbol{f}^*\in {\mathbf{H}^{-1}_D(\Omega)}$, there exist a vector field $\boldsymbol{f} \in \mathbf{L}^2(\Omega)$ and a tensor field $\mathsf{F}\in \mathbb{L}^2(\Omega)$ such that for every $\boldsymbol{v}\in \mathbf{H}^1_{D}(\Omega)$, there holds Moreover, one has that In other words, ${\mathbf{H}^{-1}_D(\Omega)}$ is isometrically isomorphic to the sum space $\mathbf{L}^2(\

Figures (5)

  • Figure 1: Square domain $\Omega$ (gray), Dirichlet boundary $\Gamma_D$ (green), and Neumann boundary $\Gamma_N$ (blue).
  • Figure 2: left: error decay plot; right: computational domain with boundary conditions.
  • Figure 3: Initial and locally refined triangulation obtained following Algorithm \ref{['alg:adaptive']} with refinement parameter $\theta=0.5$ for the Stokes minimisation problem (cf. Section \ref{['sec:stokes']}).
  • Figure 4: left: error decay plot; right: computational domain with boundary conditions.
  • Figure 5: Initial and locally refined triangulation obtained following Algorithm \ref{['alg:adaptive']} with refinement parameter $\theta=0.5$ for the Navier--Lamé minimisation minimisation problem (cf. Section \ref{['sec:stokes']}).

Theorems & Definitions (45)

  • lemma thmcounterlemma
  • proof
  • remark thmcounterremark
  • proposition thmcounterproposition: (Inverse) Marini formula
  • remark thmcounterremark
  • proof : of Proposition \ref{['prop:stokes_marini']}
  • lemma thmcounterlemma
  • proof
  • lemma thmcounterlemma
  • proof : of Lemma \ref{['lem:discrete_strong_convexity_measures_stokes']}
  • ...and 35 more