A regularization of incompressible Stokes problem with Tresca friction condition
A. Zafrar
TL;DR
This work tackles the incompressible Stokes problem with nonlinear Tresca boundary conditions by introducing a regularization $\nabla\cdot{\bf u}\in [-\epsilon,\epsilon]$ to enable near-incompressibility. It recasts the problem as a constrained elliptic variational inequality and establishes existence/uniqueness via a convergent fixed-point scheme, while also formulating a constrained minimization. An alternating direction method of multipliers (ADMM) is developed to split the minimization into tractable subproblems, with explicit updates for the velocity, divergence, and friction variables through an augmented Lagrangian. Numerical schemes (NIS and NISP) are proposed to mitigate locking and to avoid discrete inf-sup conditions, yielding a robust, finite-element-friendly framework for nearly incompressible Stokes flow with Tresca friction.
Abstract
In the present article, we introduce and study a model addressing the Stokes problem with non-linear boundary conditions of the Tresca type. We suggest a new procedure for regularizing incompressible fluid, i.e. we assume that the divergence $\nabla \cdot {\bf u}\in [-ε,\,ε]$ which leads to class of constrained elliptic variational inequalities. We use a fixed point strategy to show the existence and uniqueness of a solution and we reformulate the problem as an equivalent constrained minimization problem. An ADMM is applied to the minimization problem and some algorithm are provided.