A TWO-LEVEL DISCRETIZATION METHOD FOR THE STREAMFUNCTION FORM OF THE NAVIER-STOKES EQUATIONS
(under supervision of Prof. William Layton)
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.