Penalty method with P1/P1 finite element approximation for the Stokes equations under slip boundary condition

TL;DR

This paper introduces a penalty method combined with P1/P1 finite element approximation for solving the Stokes equations with slip boundary conditions, providing error estimates and numerical validation.

Contribution

It develops a penalty approach that simplifies implementation and improves error bounds for P1/P1 finite element solutions of the Stokes equations under slip boundary conditions.

Findings

Error estimates for velocity and pressure are established.

Numerical experiments confirm theoretical error bounds.

Improved error estimate in 2D with reduced-order integration.

Abstract

We consider the P1/P1 or P1b/P1 finite element approximations to the Stokes equations in a bounded smooth domain subject to the slip boundary condition. A penalty method is applied to address the essential boundary condition $u\cdot n = g$ on $\partial\Omega$, which avoids a variational crime and simultaneously facilitates the numerical implementation. We give $O(h^{1/2} + \epsilon^{1/2} + h/\epsilon^{1/2})$-error estimate for velocity and pressure in the energy norm, where $h$ and $\epsilon$ denote the discretization parameter and the penalty parameter, respectively. In the two-dimensional case, it is improved to $O(h + \epsilon^{1/2} + h^2/\epsilon^{1/2})$ by applying reduced-order numerical integration to the penalty term. The theoretical results are confirmed by numerical experiments.