In this work, we use a symbolic algebra package to derive a family of finite difference approximations for the biharmonic equation on a 9 point compact stencil. The solution and its first derivatives are carried as unknowns at the grid points. Dirichlet boundary conditions are thus incorporated naturally. Since the approximations use the 9 point compact stencil, no special formulas are needed near the boundaries. Both second order and fourth order discretizations are derived.
The fourth order approximations produce more accurate results than the 13 point classical stencil or the commonly used system of two second order equations coupled by the boundary condition.
The method suffers from slow convergence when classical iteration methods such as Gauss-Seidel or SOR are employed. In order to alleviate this problem we propose several multigrid techniques which exhibit grid- independent convergence and solve the biharmonic equation in a small amount of computer time. Test results from three different problems, including Stokes flow in a driven cavity, are reported.