On the application of the SCD semismooth* Newton method to solving Stokes problem with stick-slip boundary conditions

TL;DR

This paper presents a novel application of the SCD semismooth* Newton method to efficiently solve the 3D Stokes problem with complex stick-slip boundary conditions using finite element approximation.

Contribution

It introduces a new combination of variational inequality formulation with a specialized Newton method for this class of fluid dynamics problems.

Findings

The method converges reliably from arbitrary initial guesses.

Numerical experiments confirm the efficiency of the proposed approach.

The approach effectively handles the coupled inequality and equality constraints.

Abstract

The paper deals with the 3D Stokes problem with Navier-Tresca stick-slip boundary conditions. A weak formulation of this problem leads to a variational inequality of the second kind, coupled with an equality constraint. This problem is then approximated using the mixed finite element method, yielding a generalized equation, to the numerical solution of which we implement a variant of the SCD semismooth* Newton method. This includes also a globalization technique ensuring convergence for arbitrary starting points. Numerical experiments demonstrate the effeciency of this approach.