A well-posed First Order System Least Squares formulation of the instationary Stokes equations

TL;DR

This paper introduces a well-posed, space-time least squares formulation for the instationary Stokes equations, enabling efficient approximation, error estimation, and numerical validation for fluid flow problems.

Contribution

It presents a novel, well-posed first order system least squares formulation for the instationary Stokes equations with slip boundary conditions, including error estimates and numerical results.

Findings

Quasi-best approximation from finite element spaces.

Efficient evaluation of least squares norms.

Reliable a posteriori error estimator.

Abstract

In this paper, a well-posed simultaneous space-time First Order System Least Squares formulation is constructed of the instationary incompressible Stokes equations with slip boundary conditions. As a consequence of this well-posedness, the minimization over any conforming triple of finite element spaces for velocities, pressure and stress tensor gives a quasi-best approximation from that triple. The formulation is practical in the sense that all norms in the least squares functional can be efficiently evaluated. Being of least squares type, the formulation comes with an efficient and reliable a posteriori error estimator. In addition, a priori error estimates are derived, and numerical results are presented.