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

TL;DR

This paper introduces a scalable monolithic modified Newton multigrid framework for efficiently solving large, ill-conditioned nonlinear systems arising from space-time discretizations of time-dependent p-Navier-Stokes models, especially in shear-thinning regimes.

Contribution

It develops a novel surrogate tangent approach within a monolithic multigrid framework, improving conditioning and scalability for challenging nonlinear fluid flow problems.

Findings

Framework demonstrates robustness across model parameters.

Achieves scalable parallel performance.

Proves coercivity and consistency in key regimes.

Abstract

Fully implicit tensor-product space-time discretizations of time-dependent $(p,\delta)$-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 $\delta\downarrow 0$, 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 $\nu_\infty>0$ 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.