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On the domain decomposition method for the generalized Stokes problem with continuous pressure

Abstract : In this paper, the generalized Stokes problem αu−νΔu+∇p=f, ∇u=0, is considered, where ν denotes the kinematic viscosity, α is a positive parameter proportional to the inverse of the time-step, and the source term f gathers the external forcing, the nonlinear term, and the explicit part of the linear term. The system is derived from a time discretization for the Navier-Stokes equation governing the motion of a two-dimensional incompressible fluid. A domain decomposition numerical method is developed for solving the system. The original domain is divided into several nonoverlapping subdomains, and in each subdomain the concerned equations are discretized by certain finite element methods. A new feature of the method is that the pressure p is required to be piecewise linear rather than piecewise constant, so that the nonphysical oscillations in the solution using piecewise linear pressure are eliminated. However, a difficulty in using piecewise linear pressure is that the Stokes problem in each subdomain involves Dirichlet boundary conditions on pressure at the interface, which is not a real Stokes problem. To avoid the difficulty, a modification is proposed which removes the continuity conditions on the pressure across the interface. In this way, the Stokes problem in each subdomain has only Dirichlet boundary conditions on the velocity. Formulations and discussions for both the methods are presented, which shows that the modified method is simpler than the original one. Numerical experiments are displayed to show the efficiency of the methods, especially that the modified method has the same accuracy as the original one.
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Contributor : Jacques Laminie Connect in order to contact the contributor
Submitted on : Friday, September 7, 2012 - 6:17:06 PM
Last modification on : Tuesday, November 22, 2022 - 2:26:15 PM

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Jacques Laminie, Caterina Calgaro. On the domain decomposition method for the generalized Stokes problem with continuous pressure. Numerical Methods for Partial Differential Equations, 2000, 16 (1), pp.84-106. ⟨10.1002/(SICI)1098-2426(200001)16:1<84::AID-NUM7>3.0.CO;2-2⟩. ⟨hal-00730200⟩



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