# Analysis of a projection method for the Stokes problem using an $\varepsilon$-Stokes approach

## TL;DR

This paper extends the analysis of an $oldsymbol{ ext{ε}}$-Stokes projection method, connecting Stokes and pressure-Poisson problems, providing convergence, error estimates, and numerical validation for various boundary conditions.

## Contribution

It generalizes the $oldsymbol{ ext{ε}}$-Stokes approach to Neumann and mixed boundary conditions, offering new error estimates and insights into asymptotic behavior.

## Key findings

- Solutions converge to Stokes and pressure-Poisson solutions as ε varies.
- Error estimates are optimal and improved with Neumann boundary conditions.
- Numerical examples confirm theoretical error bounds and asymptotic structure.

## Abstract

We generalize pressure boundary conditions of an $\varepsilon$-Stokes problem. Our $\varepsilon$-Stokes problem connects the classical Stokes problem and the corresponding pressure-Poisson equation using one parameter $\varepsilon>0$. For the Dirichlet boundary condition, it is proven in K. Matsui and A. Muntean (2018) that the solution for the $\varepsilon$-Stokes problem converges to the one for the Stokes problem as $\varepsilon$ tends to 0, and to the one for the pressure-Poisson problem as $\varepsilon$ tends to $\infty$. Here, we extend these results to the Neumann and mixed boundary conditions. We also establish error estimates in suitable norms between the solutions to the $\varepsilon$-Stokes problem, the pressure-Poisson problem and the Stokes problem, respectively. Several numerical examples are provided to show that several such error estimates are optimal in $\varepsilon$. Our error estimates are improved if one uses the Neumann boundary conditions. In addition, we show that the solution to the $\varepsilon$-Stokes problem has a nice asymptotic structure.