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An arbitrary-order fully discrete Stokes complex on general polyhedral meshes

Marien-Lorenzo Hanot

TL;DR

The paper tackles accurate discretization of incompressible Stokes equations on general polyhedral meshes by developing an arbitrary-order fully discrete Stokes complex that augments the fully discrete de Rham framework with a complete gradient operator. It constructs five discrete spaces linked by discrete gradient, curl, and divergence operators, together with explicit interpolators and a stabilized $L^2$ product, and proves the complex is exact on contractible domains while obtaining uniform Poincaré inequalities and adjoint/primal consistency. The resulting Stokes discretization is well-posed and achieves an optimal convergence rate of $O(h^{k+1})$ for $k\,\geq 0$, under standard smoothness assumptions, with a comprehensive analysis of consistency, stability, and boundary-condition flexibility. Numerical tests in 2D and 3D across diverse polyhedral meshes corroborate the theory, demonstrating robustness and accuracy of the method for practical applications in magnetohydrodynamics and related problems.

Abstract

In this paper we present an arbitrary-order fully discrete Stokes complex on general polyhedral meshes. We enriche the fully discrete de Rham complex with the addition of a full gradient operator defined on vector fields and fitting into the complex. We show a complete set of results on the novelties of this complex: exactness properties, uniform Poincaré inequalities and primal and adjoint consistency. The Stokes complex is especially well suited for problem involving Jacobian, divergence and curl, like the Stokes problem or magnetohydrodynamic systems. The framework developed here eases the design and analysis of scheme for such problems. Schemes built that way are nonconforming and benefit from the exactness of the complex. We illustrate with the design and study of a scheme to solve the Stokes equations and validate the convergence rates with various numerical tests.

An arbitrary-order fully discrete Stokes complex on general polyhedral meshes

TL;DR

The paper tackles accurate discretization of incompressible Stokes equations on general polyhedral meshes by developing an arbitrary-order fully discrete Stokes complex that augments the fully discrete de Rham framework with a complete gradient operator. It constructs five discrete spaces linked by discrete gradient, curl, and divergence operators, together with explicit interpolators and a stabilized product, and proves the complex is exact on contractible domains while obtaining uniform Poincaré inequalities and adjoint/primal consistency. The resulting Stokes discretization is well-posed and achieves an optimal convergence rate of for , under standard smoothness assumptions, with a comprehensive analysis of consistency, stability, and boundary-condition flexibility. Numerical tests in 2D and 3D across diverse polyhedral meshes corroborate the theory, demonstrating robustness and accuracy of the method for practical applications in magnetohydrodynamics and related problems.

Abstract

In this paper we present an arbitrary-order fully discrete Stokes complex on general polyhedral meshes. We enriche the fully discrete de Rham complex with the addition of a full gradient operator defined on vector fields and fitting into the complex. We show a complete set of results on the novelties of this complex: exactness properties, uniform Poincaré inequalities and primal and adjoint consistency. The Stokes complex is especially well suited for problem involving Jacobian, divergence and curl, like the Stokes problem or magnetohydrodynamic systems. The framework developed here eases the design and analysis of scheme for such problems. Schemes built that way are nonconforming and benefit from the exactness of the complex. We illustrate with the design and study of a scheme to solve the Stokes equations and validate the convergence rates with various numerical tests.
Paper Structure (28 sections, 41 theorems, 199 equations, 27 figures)

This paper contains 28 sections, 41 theorems, 199 equations, 27 figures.

Table of Contents

  1. Introduction.
  2. Setting.
  3. Mesh and orientation.
  4. Polynomial spaces.
  5. Discrete complex.
  6. Complex definition.
  7. Interpolators.
  8. Discrete operators
  9. Gradient.
  10. Curl.
  11. Jacobian.
  12. Divergence.
  13. Discrete $L^2$-product.
  14. Results on discrete $L^2$-products.
  15. Complex property.
  16. ...and 13 more sections

Key Result

Lemma 2

Let $X$ be an element of $\mathcal{T}_h \cup \mathcal{F}_h \cup \mathcal{E}_h$. Let a positive integer $l$ and a real number $p \in [1,\infty]$ be fixed. Then, the following inequality holds: For all $v \in \mathcal{P}_{}^{l}(X)$, with hidden constant depending only on $\rho$, $l$ and $p$.

Figures (27)

  • Figure 1: Usage of the local degrees of freedom for the discrete differential operators.
  • Figure 2: "Cube" mesh.
  • Figure 3: "Prysmatic" mesh.
  • Figure 4: "Hexahedra" mesh.
  • Figure 5: "Tetgen cube" mesh.
  • ...and 22 more figures

Theorems & Definitions (94)

  • Remark 1
  • Lemma 2: Discrete inverse Poincaré
  • Lemma 3
  • Remark 4
  • Remark 5
  • Remark 6
  • Lemma 7
  • proof
  • Definition 8
  • Remark 9
  • ...and 84 more