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A Scalable Monolithic Modified Newton Multigrid Framework for Time-Dependent $p$-Navier-Stokes Flow

Nils Margenberg, Carolin Mehlmann

Abstract

Fully implicit tensor-product space-time discretizations of time-dependent $(p,δ)$-Navier-Stokes models yield, on each time step, large nonlinear monolithic saddle-point systems. In the shear-thinning regime $1<p<2$, especially as $p\downarrow 1$ and $δ\downarrow 0$, the decisive difficulty is the constitutive tangent: its ill-conditioning impairs Newton globalization and the preconditioning of the arising linear systems. We therefore develop a scalable monolithic modified Newton framework for tensor-product space-time finite elements in which the exact constitutive tangent in the Jacobian action is replaced by a better-conditioned surrogate. Picard and exact Newton serve as reference linearizations within the same algebraic framework. Scalability is achieved through matrix-free operator evaluation, a monolithic multigrid V-cycle preconditioner, order-preserving reduced Gauss-Radau time quadrature, and an inexact space-time Vanka smoother with single-time-point coefficient freezing in local patch matrices. We prove coercivity of the linearized viscous-Nitsche term in the uniformly elliptic regime $ν_\infty>0$ and consistency of the reduced time quadrature. Numerical tests demonstrate robustness with respect to model parameters, nonlinear and linear iteration counts, and scalable parallel performance.

A Scalable Monolithic Modified Newton Multigrid Framework for Time-Dependent $p$-Navier-Stokes Flow

Abstract

Fully implicit tensor-product space-time discretizations of time-dependent -Navier-Stokes models yield, on each time step, large nonlinear monolithic saddle-point systems. In the shear-thinning regime , especially as and , the decisive difficulty is the constitutive tangent: its ill-conditioning impairs Newton globalization and the preconditioning of the arising linear systems. We therefore develop a scalable monolithic modified Newton framework for tensor-product space-time finite elements in which the exact constitutive tangent in the Jacobian action is replaced by a better-conditioned surrogate. Picard and exact Newton serve as reference linearizations within the same algebraic framework. Scalability is achieved through matrix-free operator evaluation, a monolithic multigrid V-cycle preconditioner, order-preserving reduced Gauss-Radau time quadrature, and an inexact space-time Vanka smoother with single-time-point coefficient freezing in local patch matrices. We prove coercivity of the linearized viscous-Nitsche term in the uniformly elliptic regime and consistency of the reduced time quadrature. Numerical tests demonstrate robustness with respect to model parameters, nonlinear and linear iteration counts, and scalable parallel performance.

Paper Structure

This paper contains 47 sections, 7 theorems, 114 equations, 7 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Contributions and scope
  3. Related work
  4. Organization
  5. Time-dependent (p,delta)-Navier-Stokes system
  6. Problem statement and weak formulation
  7. Spaces and pairings
  8. Constitutive law
  9. Strong form
  10. Weak form
  11. Regularity and well-posedness
  12. Space-time discretization
  13. Spaces and meshes
  14. Gauss-Radau temporal nodes
  15. Semilinear spatial form and stabilization
  16. ...and 32 more sections

Key Result

Corollary 4.1

Fix a Newton state $\bm U^m$ and a tangent $\mathcal{T}_\star^m:\mathbb{R}^{d\times d}_{\mathop{\mathrm{sym}}\nolimits}\to\mathbb{R}^{d\times d}_{\mathop{\mathrm{sym}}\nolimits}$. Assume there exists $\overline\eta\ge \nu_\infty>0$ such that, for a.e. $x\in\Omega$ and all $\bm B\in\mathbb{R}^{d\time For the tangents used in this paper, these bounds hold in the uniformly elliptic regime $\nu_\infty

Figures (7)

  • Figure 1: Plots of the calculated velocity, pressure and viscosity for the convergence test with $(p,\delta)=(1.16,10^{-5})$ and $\nu=10^{-3}$ on a mesh with 262144 cells and 1024 time steps at the final time.
  • Figure 2: Performance summary. (a) Dolan-Moré profile $\pi_s(\tau)$ over the full grid (work factor $\tau$ w.r.t. the per-instance minimum). (b) $\Phi_\delta(\boldsymbol D\boldsymbol v_{\tau h})$-error versus work, $\|\Phi_\delta(\boldsymbol D\boldsymbol v)-\Phi_\delta(\boldsymbol D\boldsymbol v_{\tau h})\|_{L^2((0,T);\,L^2(\Omega))}$, on representative tuples. (c) nonlinear iteration histories $n_{NL}(n)$ for the same tuples.
  • Figure 3: Strong scaling results for the monolithic solver in degrees of freedom per second for the Picard, exact Newton, and modified Newton linearizations on two successive refinement levels.
  • Figure 4: Geometry of the test scenario with a parabolic inflow profile $\Gamma_{\textrm{in}}$, do-nothing boundary conditions at the outflow boundary $\Gamma_{\textrm{out}}$ and no-slip conditions on the obstacle and walls $\Gamma_{\textrm{wall}}$. The center of the obstacle is at $(0.2,\,0.2)$.
  • Figure 5: Plots of the calculated velocity, pressure and viscosity for the DFG benchmark with $(p,\delta)=(1.25,10^{-10})$ and $\nu=10^{-3}$ on a mesh with 4864 cells at the final time $T=8$.
  • ...and 2 more figures

Theorems & Definitions (17)

  • Remark 3.2: Global space-time form and time stepping
  • Remark 3.3: Gauss-Radau quadrature
  • Corollary 4.1: Coercivity of the linearized viscous-Nitsche term
  • proof
  • Lemma A.1: Gauss-Radau remainder
  • Lemma A.2: Smoothness and higher derivatives
  • proof
  • Lemma A.3: Quadrature error for convection and regularized stress
  • proof
  • Lemma A.4: Assembled Gauss-Radau quadrature error on $I$
  • ...and 7 more