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Achieving $h$- and $p$-robust monolithic multigrid solvers for the Stokes equations

Amin Rafiei, Scott MacLachlan

TL;DR

This work develops and analyzes monolithic geometric multigrid solvers with Vanka-type relaxation for higher-order mixed FEM discretizations of the Stokes equations, including conforming Taylor–Hood and nonconforming ${\bf H}(\text{div})$-$L^2$ schemes. By introducing topological and extended Vanka patches and comparing matrix-free (PCPATCH) versus assembled-matrix (ASMPatchPC) implementations, the authors demonstrate robust $h$- and $p$-robustness for several discretizations and reveal important stopping-criteria challenges at high polynomial orders. The study provides detailed guidance on patch design, grid-transfer strategies, and coarse-grid operators, showing promising performance on quadrilateral meshes and certain ${\bf H}(\text{div})$ discretizations, while highlighting areas where further work is needed. Overall, the results advance scalable solvers for higher-order Stokes systems and lay groundwork for extensions to Navier–Stokes and related saddle-point problems.

Abstract

The numerical analysis of higher-order mixed finite-element discretizations for saddle-point problems, such as the Stokes equations, has been well-studied in recent years. While the theory and practice of such discretizations is now well-understood, the same cannot be said for efficient preconditioners for solving the resulting linear (or linearized) systems of equations. In this work, we propose and study variants of the well-known Vanka relaxation scheme that lead to effective geometric multigrid preconditioners for both the conforming Taylor-Hood discretizations and non-conforming ${\bf H}(\text{div})$-$L^2$ discretizations of the Stokes equations. Numerical results demonstrate robust performance with respect to FGMRES iteration counts for increasing polynomial order for some of the considered discretizations, and expose open questions about stopping tolerances for effectively preconditioned iterations at high polynomial order.

Achieving $h$- and $p$-robust monolithic multigrid solvers for the Stokes equations

TL;DR

This work develops and analyzes monolithic geometric multigrid solvers with Vanka-type relaxation for higher-order mixed FEM discretizations of the Stokes equations, including conforming Taylor–Hood and nonconforming - schemes. By introducing topological and extended Vanka patches and comparing matrix-free (PCPATCH) versus assembled-matrix (ASMPatchPC) implementations, the authors demonstrate robust - and -robustness for several discretizations and reveal important stopping-criteria challenges at high polynomial orders. The study provides detailed guidance on patch design, grid-transfer strategies, and coarse-grid operators, showing promising performance on quadrilateral meshes and certain discretizations, while highlighting areas where further work is needed. Overall, the results advance scalable solvers for higher-order Stokes systems and lay groundwork for extensions to Navier–Stokes and related saddle-point problems.

Abstract

The numerical analysis of higher-order mixed finite-element discretizations for saddle-point problems, such as the Stokes equations, has been well-studied in recent years. While the theory and practice of such discretizations is now well-understood, the same cannot be said for efficient preconditioners for solving the resulting linear (or linearized) systems of equations. In this work, we propose and study variants of the well-known Vanka relaxation scheme that lead to effective geometric multigrid preconditioners for both the conforming Taylor-Hood discretizations and non-conforming - discretizations of the Stokes equations. Numerical results demonstrate robust performance with respect to FGMRES iteration counts for increasing polynomial order for some of the considered discretizations, and expose open questions about stopping tolerances for effectively preconditioned iterations at high polynomial order.
Paper Structure (11 sections, 2 theorems, 34 equations, 15 figures, 1 algorithm)

This paper contains 11 sections, 2 theorems, 34 equations, 15 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Mixed finite-element discretization of Stokes equations
  3. Stable nonconforming discretizations
  4. Monolithic multigrid solvers
  5. Space decomposition and patch-based relaxation
  6. Numerical Experiments
  7. Taylor-Hood on triangular grids
  8. Taylor-Hood on quadrilateral grids
  9. ${\bf H}(\text{div})$-$L^2$ discretizations on triangular grids
  10. Stopping criteria
  11. Conclusion and future research direction

Key Result

Theorem 2.1

Let $({\bf u},p)\in {\bf H_0^1}(\Omega)\times L_0^2(\Omega)$ be the solution of eq:VariationalStokes1 and suppose that Then, there exists a unique solution $({\bf u}_h,p_h)\in ({\bf V}_h,Q_h)$ of eq:VariationalDiscStokes, and the following error bounds are satisfied where $\gamma$ is the coercivity constant of $a({\bf u},{\bf v})$ and the approximation errors, $E_{\bf u}$ and $E_p$, are defined

Figures (15)

  • Figure 1: $h \mkern 2mu$-multigrid coarsening and refinement for $({\bf Q_4},Q_3)$ ( left), $({\bf P_4},P_3)$ ( center) and $(\mathcal{BDM}_3,{dP_2})$ ( right) discretizations of the Stokes equations. Big (green) and small (black) circles are associated with velocity and pressure DoFs, respectively.
  • Figure 2: Left: Topological Vanka patches for $({\bf P_4},P_3)$ Taylor-Hood discretization, showing nodal (left), edge (right) and cell (top) patches for composite Vanka relaxation scheme. Right: Topological Vanka patches for $({\bf Q_4},Q_3)$ Taylor-Hood discretization, showing nodal (left), edge (right) and cell (top) patches for composite Vanka relaxation scheme. Big (green) and small (black) circles are associated with velocity and pressure DoFs, respectively. Within each patch, we show only the included degrees of freedom, suppressing pressure DoFs that are not included in the patches.
  • Figure 3: Left: Extended Vanka patches for $(\mathcal{ BDM}_3,{dP}_2)$. Right: Extended Vanka patches for $(\mathcal{RT}_3,{dP}_2)$. Big (green) and small (black) circles are associated with velocity and pressure DoFs, respectively. Within each patch, we show only the included degrees of freedom, suppressing pressure DoFs that are not included in the patches.
  • Figure 4: Comparison of iteration counts ( top row) and time-to-solution ( bottom row) for the Taylor-Hood solvers on triangular grids with order $k=7$. Results at left use ASMPatchPC for the Vanka relaxation, while those at right use PCPATCH.
  • Figure 5: Comparison of time-to-solution for two implementations of solvers for the Taylor-Hood discretization on triangular grids. At left, we use ASMPatchPC for relaxation. At right, we use PCPATCH for relaxation.
  • ...and 10 more figures

Theorems & Definitions (3)

  • Theorem 2.1
  • Definition 2.1
  • Theorem 2.2