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On pressure boundary conditions for the incompressible Navier-Stokes equations. (English) Zbl 0644.76025

The authors have very successfully analysed various situations where the boundary conditions for the pressure Poisson equation would lead to discrepancies. They have attempted to settle the issue (1) via the continuum partial differential equations, (2) via the analysis of several consistent discretized approximations to the PDEs and (3) by numerical examples.
The following observations are noted. (i) To solve the continuum Poisson equation for the pressure only the Neumann boundary condition is always appropriate; i.e., it provides a unique solution to \(t\geq 0\). The Dirichlet boundary condition is generally only appropriate for \(t>0\); it often does not apply at \(t=0\). The unique solution obtained using either boundary condition will, for \(t>0\), satisfy the often boundary condition provided the Neumann boundary condition is applied at \(t=0\). (ii) Any consistent discrete approximation of the original (primitive) equations contains, as an automatic built-in boundary condition for the (implied) discrete pressure poisson equation, the Neumann boundary condition; for \(t\geq 0\). It does not obviously satisfy the Dirichlet boundary condition. (iii) The converged numerical solution from (ii) will, however, also satisfy the Dirichlet boundary condition owing to (i); but in general for \(t>0\). These observations have proper basis for all numerical approximation methods and the authors hope to extend this work in other directions where computational difficulties often occur.
Reviewer: V.Subba Rao

MSC:

76D05 Navier-Stokes equations for incompressible viscous fluids
35Q99 Partial differential equations of mathematical physics and other areas of application
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