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A finite element analysis of a linearized problem of the Navier-Stokes equations with surface tension. (English) Zbl 0993.76048

The authors discuss applications of finite element methods to numerical solution of a boundary value problem for linearized stationary Navier-Stokes equations. The domain is a rectangle, and periodicity conditions are given with respect to \(x_1\). For \(x_2\)-direction, the Dirichlet conditions are given on the lower side; on the upper side, the conditions correspond to the presence of surface tension and contain second-order derivatives along the side. By this reason, the operator formulation should be given in a nonstandard Hilbert space which (for velocity) is known as strengthened Sobolev space. Also is of importance that the space for pressure is \(L_2(\Omega)\), and that the linearized problem is not symmetric. The authors present an investigation of certain finite element methods (based on triangulation of \(\overline\Omega\)) that takes into account the above complications. They discuss also a model numerical example. It should be noted that the strengthened Sobolev spaces have been considered in many papers – the references can be found in [E. G. D’yakonov, Optimization in solving elliptic problems. Boca Raton, FL: CRC Press. xxviii (1996; Zbl 0852.65087)]. Also, some results on numerical methods for similar operator problems can be found in chapters 7 and 8 of the cited book.

MSC:

76M10 Finite element methods applied to problems in fluid mechanics
76D05 Navier-Stokes equations for incompressible viscous fluids
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
76D45 Capillarity (surface tension) for incompressible viscous fluids

Citations:

Zbl 0852.65087
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