Numerical solution of differential and integral equations
The solution of the Navier Ц Stokes equations for incompressible fluid flow is one of the difficult problems in the field of computational fluid dynamics. The Navier Ц Stokes equations form a set of coupled equations for both velocity and pressure (the gradient of the pressure). One of the main problems related to the numerical solution of these equations is the imposition of the incompressibility constraint and the calculation of the pressure. In this report the finite element methods (the Uzawa algorithm and the mixed method) are considered. Convergence of the finite element solution depends on the choice of trial and interpolation spaces for the velocity and the pressure. The interpolation functions for the velocity and the pressure must satisfy Ladyzhenskaya Ц Babuska Ц Brezzi condition (LBB). For approximation of the velocity and the pressure the interpolation functions are chosen. Linearization of the convective terms are performed by using the Picard iteration scheme. The projection method using equal Ц order basis functions for the velocity and the pressure is considered. The projection method is performed on the continuous form of the equations, yielding a set of decoupled equations that do not have the form of a saddle Ц point problem anymore and thus avoid the need to satisfy the LBB. The properties of the schemes are tested by means of an analytical test examples.
Note. Abstracts are published in author's edition
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