A Dual-Mixed Approximation for a Huber Regularization of Generalized $p$-Stokes Viscoplastic Flow Problems

TL;DR

This paper introduces a dual-mixed finite element approach with a Huber regularization for modeling viscoplastic flows, providing a convergent numerical scheme and demonstrating its effectiveness through experiments.

Contribution

It develops a novel dual-mixed formulation with Huber regularization for viscoplastic flow problems, including a semismooth Newton solver with proven local convergence.

Findings

The proposed scheme is stable and convergent.

Numerical experiments confirm the method's efficiency in 2D and 3D.

The approach effectively handles yield stress in flow simulations.

Abstract

In this paper, we propose a dual-mixed formulation for stationary viscoplastic flows with yield, such as the Bingham or the Herschel-Bulkley flow. The approach is based on a Huber regularization of the viscosity term and a two-fold saddle point nonlinear operator equation for the resulting weak formulation. We provide the uniqueness of solutions for the continuous formulation and propose a discrete scheme based on Arnold-Falk-Winther finite elements. The discretization scheme yields a system of slantly differentiable nonlinear equations, for which a semismooth Newton algorithm is proposed and implemented. Local superlinear convergence of the method is also proved. Finally, we perform several numerical experiments in two and three dimensions to investigate the behavior and efficiency of the method.