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Stabilised finite element method for Stokes problem with nonlinear slip condition

Tom Gustafsson, Juha Videman

TL;DR

This work addresses Stokes flow with a nonlinear slip boundary condition of friction type (Tresca) by formulating it as a mixed variational inequality using a Lagrange multiplier $\boldsymbol{\lambda}$. A residual-stabilised finite element method is developed to achieve uniform stability for $(\boldsymbol{u}, p, \boldsymbol{\lambda})$, enabling low-order discretisations and a rigorous a priori error framework. An Uzawa-type solver with an explicit projection enforces the tangential bound on the multiplier, and the authors prove discrete stability, quasi-optimality, and, under mild regularity, optimal convergence rates for the lowest-order scheme. Numerical experiments on square and curved domains validate linear convergence for the velocity and pressure and demonstrate favorable behavior of the Lagrange multiplier, highlighting the practical applicability of the method to frictional boundary problems in fluid and solid mechanics.

Abstract

This work introduces a stabilised finite element formulation for the Stokes flow problem with a nonlinear slip boundary condition of friction type. The boundary condition is enforced with the help of an additional Lagrange multiplier and the stabilised formulation is based on simultaneously stabilising both the pressure and the Lagrange multiplier. We establish the stability and the a priori error analyses, and perform a numerical convergence study in order to verify the theory.

Stabilised finite element method for Stokes problem with nonlinear slip condition

TL;DR

This work addresses Stokes flow with a nonlinear slip boundary condition of friction type (Tresca) by formulating it as a mixed variational inequality using a Lagrange multiplier . A residual-stabilised finite element method is developed to achieve uniform stability for , enabling low-order discretisations and a rigorous a priori error framework. An Uzawa-type solver with an explicit projection enforces the tangential bound on the multiplier, and the authors prove discrete stability, quasi-optimality, and, under mild regularity, optimal convergence rates for the lowest-order scheme. Numerical experiments on square and curved domains validate linear convergence for the velocity and pressure and demonstrate favorable behavior of the Lagrange multiplier, highlighting the practical applicability of the method to frictional boundary problems in fluid and solid mechanics.

Abstract

This work introduces a stabilised finite element formulation for the Stokes flow problem with a nonlinear slip boundary condition of friction type. The boundary condition is enforced with the help of an additional Lagrange multiplier and the stabilised formulation is based on simultaneously stabilising both the pressure and the Lagrange multiplier. We establish the stability and the a priori error analyses, and perform a numerical convergence study in order to verify the theory.
Paper Structure (10 sections, 7 theorems, 71 equations, 8 figures)

This paper contains 10 sections, 7 theorems, 71 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Strong formulation
  3. Mixed variational formulation
  4. Stabilised finite element method
  5. Error analysis
  6. Solution algorithm
  7. Numerical experiment
  8. Convergence study
  9. Curved boundary
  10. Conclusions

Key Result

Theorem 1

For every $(\boldsymbol{w}, r, \boldsymbol{\xi}) \in \boldsymbol{V} \times Q \times \boldsymbol{\varLambda}$ there exists $(\boldsymbol{v}, q, \boldsymbol{\mu}) \in \boldsymbol{V} \times Q \times \boldsymbol{\varLambda}$ satisfying and

Figures (8)

  • Figure 1: Some meshes from the uniform sequence.
  • Figure 2: The velocity magnitude (top) and the pressure (bottom) computed using the finest mesh in the uniform sequence.
  • Figure 3: The components of the discrete Lagrange multiplier solution at the boundary $x=1$ plotted for four different meshes from coarsest (top) to finest (bottom).
  • Figure 4: The convergence of the error in the velocity.
  • Figure 5: The convergence of the error in the pressure.
  • ...and 3 more figures

Theorems & Definitions (13)

  • Remark 1
  • Remark 2
  • Theorem 1: Continuous stability
  • Lemma 1: Inverse estimates
  • Lemma 2: Discrete trace estimate
  • Theorem 2: Discrete stability
  • proof
  • Lemma 3: Lower bound for the boundary residual
  • Lemma 4: Lower bound for the interior residual
  • Theorem 3: Quasi-optimality
  • ...and 3 more