**A TWO-LEVEL DISCRETIZATION METHOD FOR THE
STREAMFUNCTION FORM OF THE NAVIER-STOKES EQUATIONS**

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# Faisal Abdul-Karim Fairag,PhD

(under supervision of
**Prof.
William Layton**)

## University of Pittsburgh, 1998

ABSTRACT : We analyze a two-level
finite element method for the streamfunction formulation of the Navier-Stokes
equations. This report presents the two-level algorithm and a priori error
analysis for the case of conforming elements. The streamfunction formulation in
two dimensions has the great advantages that one solves for only a single
scalar variable rather than a coupled system. Further, the incompressible
constraint is automatically satisfied so there are no compatibility conditions
between velocity and pressure spaces. The disadvantage is that the linear
system, though small, arises from a fourth order problem so it can be very ill
conditioned. The nonlinear system is also, at higher Reynolds numbers, very
sensitive to small perturbations. The two-level algorithm consists of solving a
small nonlinear system on the coarse mesh, then solving a large linear system
on the fine mesh. The basic results states that the error between the coarse
and fine meshes are related superlinearly.

FORTRAN
programs for this algorithm and a complete discussion of these programs are
included. These programs are used to solve the Navier-Stokes equations with a
known solution in a rectangular domain for a range of Reynolds numbers to
compare one level vs. two level methods in terms of computer time. Also, we
solve the driven flow in a square cavity. These flows have been widely used as
test cases for validating incompressible fluid dynamics algorithms.
Streamfunction contours are displayed showing the main features of the flow.

A
posteriori error estimator for the two-level algorithm is derived which can be
used as an indicator for an assessment of the reliability of the results.