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The finite element method over a simple stabilizing grid applied to fluid flow problems

Aydın, Selçuk Han
We consider the stabilized finite element method for solving the incompressible Navier-Stokes equations and the magnetohydrodynamic (MHD) equations in two dimensions. The well-known instabilities arising from the application of standard Galerkin finite element method are eliminated by using the stabilizing subgrid method (SSM), the streamline upwind Petrov-Galerkin (SUPG) method, and the two-level finite element method (TLFEM). The domain is discretized into a set of regular triangular elements. In SSM, the finite-dimensional spaces employed consist of piecewise continuous linear interpolants enriched with the residual-free bubble functions. To find the bubble part of the solution, a two-level finite element method with a stabilizing subgrid of a single node is described and its applications to the Navier-Stokes equations and MHD equations are displayed. This constitutes the main original contribution of this thesis. Numerical approximations employing the proposed algorithms are presented for some benchmark problems. The results show that the proper choice of the subgrid node is crucial to get stable and accurate numerical approximations consistent with the physical configuration of the problem at a cheap computational cost. The stabilized finite element method of SUPG type is applied to the unsteady Navier-Stokes equations together with a finite element discretization in the time domain. Thus, oscillations in the solution and the need of very small time increment are avoided in obtaining stable solutions.