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arXiv · 1812.10250

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

Masato Kimura·Kazunori Matsui·Adrian Muntean·Hirofumi Notsu
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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.

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Masato Kimura, Kazunori Matsui, Adrian Muntean, Hirofumi Notsu. 2018-12-26. Analysis of a projection method for the Stokes problem using an $\varepsilon$-Stokes approach. https://arxiv.org/abs/1812.10250

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