Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A pressure-robust Discrete de Rham scheme for the Navier-Stokes equations

Daniele A. Di Pietro, Jerome Droniou, Jia Jia Qian

Abstract

In this work we design and analyse a Discrete de Rham (DDR) method for the incompressible Navier-Stokes equations. Our focus is, more specifically, on the SDDR variant, where a reduction in the number of unknowns is obtained using serendipity techniques. The main features of the DDR approach are the support of general meshes and arbitrary approximation orders. The method we develop is based on the curl-curl formulation of the momentum equation and, through compatibility with the Helmholtz-Hodge decomposition, delivers pressure-robust error estimates for the velocity. It also enables non-standard boundary conditions, such as imposing the value of the pressure on the boundary. In-depth numerical validation on a complete panel of tests including general polyhedral meshes is provided. The paper also contains an appendix where bounds on DDR potential reconstructions and differential operators are proved in the more general framework of Polytopal Exterior Calculus.

A pressure-robust Discrete de Rham scheme for the Navier-Stokes equations

Abstract

In this work we design and analyse a Discrete de Rham (DDR) method for the incompressible Navier-Stokes equations. Our focus is, more specifically, on the SDDR variant, where a reduction in the number of unknowns is obtained using serendipity techniques. The main features of the DDR approach are the support of general meshes and arbitrary approximation orders. The method we develop is based on the curl-curl formulation of the momentum equation and, through compatibility with the Helmholtz-Hodge decomposition, delivers pressure-robust error estimates for the velocity. It also enables non-standard boundary conditions, such as imposing the value of the pressure on the boundary. In-depth numerical validation on a complete panel of tests including general polyhedral meshes is provided. The paper also contains an appendix where bounds on DDR potential reconstructions and differential operators are proved in the more general framework of Polytopal Exterior Calculus.
Paper Structure (26 sections, 9 theorems, 113 equations, 14 figures)

This paper contains 26 sections, 9 theorems, 113 equations, 14 figures.

Table of Contents

  1. Introduction
  2. Continuous setting
  3. Numerical scheme and main result
  4. Mesh and notation
  5. Polynomial spaces
  6. Serendipity Discrete de Rham complex
  7. SDDR spaces and serendipity operators
  8. Gradient space
  9. Curl space
  10. Divergence space
  11. Discrete $L^2$-products
  12. SDDR scheme for the Navier--Stokes equations
  13. Relevant constants
  14. Discrete norms
  15. Error estimates
  16. ...and 11 more sections

Key Result

Theorem 2

Assume that eq:weak has a solution $(\boldsymbol{u},p)\in\boldsymbol{H}(\mathop{\mathrm{\bf curl}}\nolimits;\Omega)\times H^1(\Omega)$ such that We denote by the part of $\boldsymbol{f}$ depending only on the velocity. Under the mesh assumption of Section sec:mesh, suppose further that Then, if $(\underline{\boldsymbol{u}}_h,\underline{p}_h)$ is the solution to the SDDR scheme eq:ddr, the follo

Figures (14)

  • Figure 1: Errors on $\boldsymbol{u}$, $\lambda=1$
  • Figure 2: Errors on $\boldsymbol{u}$, $\lambda=10^2$
  • Figure 3: Errors on $\nabla p$, $\lambda=1$
  • Figure 4: Errors on $\nabla p$, $\lambda=10^2$
  • Figure 6: Errors on $\boldsymbol{u}$, $\lambda=1$
  • ...and 9 more figures

Theorems & Definitions (21)

  • Remark 1: Discrete Sobolev embedding
  • Theorem 2: Error estimate for the SDDR scheme
  • Remark 3: Data smallness assumption
  • Lemma 4: A priori estimate on the velocity
  • proof
  • Lemma 5: A priori estimates on the pressure
  • proof
  • Lemma 6: Consistency bounds
  • proof
  • Remark 7: Mixed boundary conditions
  • ...and 11 more