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Fast solution of incompressible flow problems with two-level pressure approximation

Jennifer Pestana, David J. Silvester

TL;DR

This work addresses efficient linear solvers for incompressible flow discretizations that augment the usual Taylor–Hood pressure space with a piecewise constant component to enforce local mass conservation. The authors analyze the resulting two-field pressure mass matrix $M_Q$, show its singularity under frame-based pressure representations, and develop preconditioners for the resulting saddle-point systems, including a robust two-stage pressure-convection-diffusion (PCD) strategy and a least-squares commutator (LSC) variant for Oseen flow. They demonstrate that standard diagonalisations of the $M_Q$ block are inadequate and that EST-MINRES provides reliable estimates of the discrete inf--sup constant $\gamma$, enabling mesh-independent convergence in many cases. The practical impact is a framework for efficiently solving enriched Taylor–Hood systems with local mass conservation, yielding scalable, accurate solutions and enabling better computation of derived quantities such as wall shear stress; code implementing these strategies is publicly available.

Abstract

This paper develops efficient preconditioned iterative solvers for incompressible flow problems discretised by an enriched Taylor-Hood mixed approximation, in which the usual pressure space is augmented by a piecewise constant pressure to ensure local mass conservation. This enrichment process causes over-specification of the pressure when the pressure space is defined by the union of standard Taylor-Hood basis functions and piecewise constant pressure basis functions, which complicates the design and implementation of efficient solvers for the resulting linear systems. We first describe the impact of this choice of pressure space specification on the matrices involved. Next, we show how to recover effective solvers for Stokes problems, with preconditioners based on the singular pressure mass matrix, and for Oseen systems arising from linearised Navier-Stokes equations, by using a two-stage pressure convection-diffusion strategy. The codes used to generate the numerical results are available online.

Fast solution of incompressible flow problems with two-level pressure approximation

TL;DR

This work addresses efficient linear solvers for incompressible flow discretizations that augment the usual Taylor–Hood pressure space with a piecewise constant component to enforce local mass conservation. The authors analyze the resulting two-field pressure mass matrix , show its singularity under frame-based pressure representations, and develop preconditioners for the resulting saddle-point systems, including a robust two-stage pressure-convection-diffusion (PCD) strategy and a least-squares commutator (LSC) variant for Oseen flow. They demonstrate that standard diagonalisations of the block are inadequate and that EST-MINRES provides reliable estimates of the discrete inf--sup constant , enabling mesh-independent convergence in many cases. The practical impact is a framework for efficiently solving enriched Taylor–Hood systems with local mass conservation, yielding scalable, accurate solutions and enabling better computation of derived quantities such as wall shear stress; code implementing these strategies is publicly available.

Abstract

This paper develops efficient preconditioned iterative solvers for incompressible flow problems discretised by an enriched Taylor-Hood mixed approximation, in which the usual pressure space is augmented by a piecewise constant pressure to ensure local mass conservation. This enrichment process causes over-specification of the pressure when the pressure space is defined by the union of standard Taylor-Hood basis functions and piecewise constant pressure basis functions, which complicates the design and implementation of efficient solvers for the resulting linear systems. We first describe the impact of this choice of pressure space specification on the matrices involved. Next, we show how to recover effective solvers for Stokes problems, with preconditioners based on the singular pressure mass matrix, and for Oseen systems arising from linearised Navier-Stokes equations, by using a two-stage pressure convection-diffusion strategy. The codes used to generate the numerical results are available online.
Paper Structure (10 sections, 2 theorems, 45 equations, 9 figures, 4 tables, 1 algorithm)

This paper contains 10 sections, 2 theorems, 45 equations, 9 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Two-field pressure mass matrix
  3. Augmented Taylor--Hood Elements
  4. Preconditioning considerations
  5. Approximating the two-level pressure mass matrix
  6. Reliable computation of the discrete inf--sup constant
  7. Two-field pressure preconditioning strategies for Oseen flow.
  8. Pressure convection-diffusion preconditioning for Oseen flow.
  9. Least square commutator approximation for Oseen flow
  10. Conclusions

Key Result

Proposition 1

Let be the pressure mass matrix for the enriched Taylor--Hood pressure space $Q_{h}^{\star}$ in eq:eth_pressure_space, specified by the frame eq:eth_spec. Then, $\mathop{\mathrm{null}}\limits(M_Q) = \mathop{\mathrm{span}}\limits\{\boldsymbol{k}\}$, where $\boldsymbol{k}$ is given in eq:pressure_null_vec

Figures (9)

  • Figure 1: Representative $\mathbb{P}_2$--$\mathbb{P}_1^*$ pressure field solution for the cavity flow in Example \ref{['ex:2d_cavity']} computed on a uniform mesh with 2048 right-angled triangles and 1089 vertices. Contours are equally spaced between the maximum and minimum values in both plots.
  • Figure 2: EST-MINRES estimates of the discrete inf--sup constant $\gamma^2$ at each iteration for Example \ref{['ex:2d_cavity']} and preconditioner $\mathcal{P}_1$ with $\mathbb{P}_2$--$\mathbb{P}_1$ (left) and $\mathbb{P}_2$--$\mathbb{P}_1^\ast$ (right) elements for the grids specified in Table \ref{['tab:2d_infsup']}.
  • Figure 3: EST-MINRES estimates of the discrete inf--sup constant $\gamma^2$ at each iteration for Example \ref{['ex:3d_cavity']} and preconditioner $\mathcal{P}_1$ with $\hbox{Q}_2$--$\hbox{Q}_1$ (left) and $\hbox{Q}_2$--$\hbox{Q}_1^\ast$ (right) elements for the grids specified in Table \ref{['tab:3d_infsup']}.
  • Figure 4: Representative $\mathbb{P}_2$--$\mathbb{P}_1^*$ pressure field solution for flow over a step ($\mathcal{R} = 100$) computed on a uniform mesh with 5632 right-angled triangles and 2945 vertices.
  • Figure 5: Comparison of divergence error and $\mathbb{P}_2$--$\mathbb{P}_1^*$ centroid pressure solution for cavity flow ($\mathcal{R}=200$) computed on a uniform mesh with 2048 right-angled triangles and 1089 vertices.
  • ...and 4 more figures

Theorems & Definitions (5)

  • Proposition 1
  • proof
  • Remark 2
  • Proposition 4
  • proof