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Arbitrary-order pressure-robust DDR and VEM methods for the Stokes problem on polyhedral meshes

Lourenço Beirão da Veiga, Franco Dassi, Daniele A. Di Pietro, Jérôme Droniou

TL;DR

The paper advances pressure-robust discretizations for the Stokes problem on polyhedral meshes by developing high-order DDR and VEM schemes. Both approaches rely on a compatible discrete de Rham framework and a projection of the forcing onto discrete $\boldsymbol{H}(\mathrm{curl})$ spaces, combined with commuting diagrams that ensure the irrotational portion of the load does not degrade velocity accuracy. They prove $h^{k+1}$-type velocity-error bounds under suitable regularity and verify these results with 3D numerical tests demonstrating robustness to large irrotational forcing components. A key contribution is the explicit bridging of DDR and VEM perspectives, showing how one framework can be reformulated in the other and how their DOF maps commute, enabling cross-fertilization and a unified view of polyhedral discretizations for incompressible flows.

Abstract

This paper contains two major contributions. First we derive, following the discrete de Rham (DDR) and Virtual Element (VEM) paradigms, pressure-robust methods for the Stokes equations that support arbitrary orders and polyhedral meshes. Unlike other methods presented in the literature, pressure-robustness is achieved here without resorting to an $\boldsymbol{H}({\rm div})$-conforming construction on a submesh, but rather projecting the volumetric force onto the discrete $\boldsymbol{H}({\bf curl})$ space. The cancellation of the pressure error contribution stems from key commutation properties of the underlying DDR and VEM complexes. The pressure-robust error estimates in $h^{k+1}$ (with $h$ denoting the meshsize and $k\ge 0$ the polynomial degree of the DDR or VEM complex) are proven theoretically and supported by a panel of three-dimensional numerical tests. The second major contribution of the paper is an in-depth study of the relations between the DDR and VEM approaches. We show, in particular, that a complex developed following one paradigm admits a reformulation in the other, and that couples of related DDR and VEM complexes satisfy commuting diagram properties with the degrees of freedom maps.

Arbitrary-order pressure-robust DDR and VEM methods for the Stokes problem on polyhedral meshes

TL;DR

The paper advances pressure-robust discretizations for the Stokes problem on polyhedral meshes by developing high-order DDR and VEM schemes. Both approaches rely on a compatible discrete de Rham framework and a projection of the forcing onto discrete spaces, combined with commuting diagrams that ensure the irrotational portion of the load does not degrade velocity accuracy. They prove -type velocity-error bounds under suitable regularity and verify these results with 3D numerical tests demonstrating robustness to large irrotational forcing components. A key contribution is the explicit bridging of DDR and VEM perspectives, showing how one framework can be reformulated in the other and how their DOF maps commute, enabling cross-fertilization and a unified view of polyhedral discretizations for incompressible flows.

Abstract

This paper contains two major contributions. First we derive, following the discrete de Rham (DDR) and Virtual Element (VEM) paradigms, pressure-robust methods for the Stokes equations that support arbitrary orders and polyhedral meshes. Unlike other methods presented in the literature, pressure-robustness is achieved here without resorting to an -conforming construction on a submesh, but rather projecting the volumetric force onto the discrete space. The cancellation of the pressure error contribution stems from key commutation properties of the underlying DDR and VEM complexes. The pressure-robust error estimates in (with denoting the meshsize and the polynomial degree of the DDR or VEM complex) are proven theoretically and supported by a panel of three-dimensional numerical tests. The second major contribution of the paper is an in-depth study of the relations between the DDR and VEM approaches. We show, in particular, that a complex developed following one paradigm admits a reformulation in the other, and that couples of related DDR and VEM complexes satisfy commuting diagram properties with the degrees of freedom maps.
Paper Structure (49 sections, 9 theorems, 135 equations, 18 figures, 1 table)

This paper contains 49 sections, 9 theorems, 135 equations, 18 figures, 1 table.

Table of Contents

  1. Introduction
  2. Setting
  3. Mesh and orientation of mesh entities
  4. Differential operators on faces and tangential trace
  5. Lebesgue and Hilbert spaces
  6. Polynomial spaces and decompositions
  7. DDR scheme
  8. Spaces
  9. Discrete vector calculus operators
  10. Curl
  11. Gradient
  12. Discrete potentials and $L^2$-products
  13. Discrete problem and convergence
  14. VEM scheme
  15. Local spaces on faces
  16. ...and 34 more sections

Key Result

Theorem 4

Denote by $\boldsymbol{u}\in\boldsymbol{H}(\mathop{\mathrm{\bf curl}}\nolimits;\Omega)\cap\boldsymbol{H}(\mathop{\mathrm{div}}\nolimits;\Omega)$ and $p\in H^1(\Omega)\cap L^2_0(\Omega)$, respectively, the velocity and pressure fields solution of the weak formulation eq:variational, and by $\underlin

Figures (18)

  • Figure 1: Tetrahedral mesh
  • Figure 2: Voronoi mesh
  • Figure 4: DDR scheme errors on $\boldsymbol{u}$
  • Figure 5: DDR scheme errors on $p$
  • Figure 6: VEM scheme errors on $\boldsymbol{u}$
  • ...and 13 more figures

Theorems & Definitions (28)

  • Remark 1: Domain of $\underline{\boldsymbol{I}}_{\mathop{\mathrm{\bf curl}}\nolimits,h}^k$
  • Remark 2: Stabilisation
  • Remark 3: Discretisation of the volumetric force term
  • Theorem 4: Error estimate for the DDR scheme \ref{['eq:discrete']}
  • proof
  • Remark 5: Pressure robustness
  • Remark 6: Edge serendipity operator
  • Remark 7: Alternative stabilisation
  • Remark 8: Alternative choices of degrees of freedom
  • Remark 9: Enhancement
  • ...and 18 more