A variational finite element method for the two-dimensional Stokes problem
In two dimensions, the steady Stokes problem in stream function-vorticity variables is reduced to a Dirichlet problem for the biharmonic operator. A new formulation for the decomposition of the biharmonic operator using the potential theory is presented. The problem comes down to the calculation of a function defined on the boundary, which is the solution of a variational problem. The first part of the paper is devoted to the analysis of the continuous problem, and the variational formulation on the boundary is studied. By using a suitable finite element approximation, similar properties for the discrete problem are found. These properties imply that the discrete function is the solution of a linear system, whose matrix is full, but symmetric and positive-definite. Among the iterative methods, we use the conjugate gradient method, which seems suitable in the context. Numerical results concerning the problem of steady flow in a square cavity are presented.
scientific article; zbMATH DE number 873988