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Siberian Journal of Numerical Mathematics

2017 year, number 4

A boundary value problem for one overdetermined stationary system emerging in the two-velocity hydrodynamics

M.V. Urev1,2,3, Kh.Kh. Imomnazarov1, Jian-Gang Tang4
1Institute of Computational Mathematics and Mathematical Geophysics SB RAS, pr. Acad. Lavrentieva 6, Novosibirsk, 630090, Russia
2Novosibirsk State University, Pirogova st., 2, Novosibirsk, 630090, Russia
3Siberian Institute of Management, ul. Nizhegorodskya, 6, Novosibirsk, 630102
4YiLi Normal University, 448, Jiefang Road, Yinning Xinjiang, P.R. of China
Keywords: переопределенная стационарная система двухскоростной гидродинамики, множитель Лагранжа, метод конечных элементов, overdetermined two-velocity stationary hydrodynamics system, Lagrange multiplier, finite element method

Abstract

In this paper we investigate the two-velocity stationary hydrodynamics system with a single pressure and inhomogeneous divergent and boundary conditions for the two velocities. This system is overdetermined. By replacing the unknown functions, the problem is reduced to a homogeneous one. The solution of the resulting system is reduced to the consecutive solutions of the two boundary value problems: the Stokes problem for a single velocity and pressure, and overdetermined system for the other velocity. We present the generalized statements of these problems and their discrete approximation using the finite element method. To solve the overdetermined problem we apply a version of the regularization methods.