摩擦型の境界条件のもとでのストークス流に対するUzawaの方法について

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This paper is concerned with the Uzawa algorithm applied to the stationary Stokes flow problem which is formulated as a boundary value problem for the Stokes equation with the equation of continuity and the boundary condition. In the spirit of the Uzawa algorithm which originated from the optimization problem with linear constraints, we regard the equation of continuity as the linear constraints and regard the pressure as the corresponding Lagrange multiplier. Namely, we have to go to the variational formulation of the Stokes boundary value problem. If the boundary condition is of the second type on a part S of the boundary, it is incorporated as a kind of natural boundary condition in terminology of variational method. Then we state the Uzawa algorithm to generate approximate sequences for the flow velocity u and p. The convergence of the approximation to the exact solution with error decaying exponentially, namely, the exponential convergence is proved which is new to the case with boundary condition of the second type. We then proceed to the consideration of the algorithm applied to the above-mentioned boundary problem with the boundary condition replaced (at a part of the boundary) by the so-called boundary condition of frictional type. This boundary condition reduces to the usual homogeneous Dirichlet condition for the velocity if the magnitude of the stress is below a threshold, while it admits of some leak or slip of the liquid on the boundary. We deal with this non-linear boundary condition and propose a kind of the Uzawa algorithm and clarify by numerical experiments that the Uzawa algorithm is nicely applied. In a forthcoming paper we intend to prove or disprove the exponential convergence of the algorithm.
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摩擦型の境界条件のもとでのストークス流に対するUzawaの方法について

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