On a finite element approximation of the Stokes problem under leak or slip boundary conditions of friction type

TL;DR

This paper develops a finite element method for the Stokes problem with nonlinear slip or leak boundary conditions, providing theoretical analysis and numerical validation of the approach.

Contribution

It introduces a variational inequality formulation for the problem, proves existence and uniqueness, and offers an error estimate along with a numerical solution method.

Findings

The method achieves accurate approximations under nonlinear boundary conditions.

The Uzawa-type iterative method effectively solves the discretized problem.

Numerical examples confirm the theoretical error estimates.

Abstract

A finite element approximation of the Stokes equations under a certain nonlinear boundary condition, namely, the slip or leak boundary condition of friction type, is considered. We propose an approximate problem formulated by a variational inequality, prove an existence and uniqueness result, present an error estimate, and discuss a numerical realization using an iterative Uzawa-type method. Several numerical examples are provided to support our theoretical results.