Discretization Error Estimates for Certain Splitting Procedures for Solving First Biharmonic Boundary Value Problems

This paper considers the Dirichlet problem for the two-dimensional biharmonic equation in a bounded region consisting of a finite sum of rectangles. The biharmonic equation is first split into two Poisson equations and two classes of finite difference schemes are defined for obtaining the numerical solution. These classes correspond to the type of difference approximation defined for the missing boundary condition. Discretization error for the difference schemes in these two classes is shown to be of order $h^{{3 / 2}} $ and $h^2 $, respectively, as the mesh size $h \to 0$. The effect on discretization error of the different approximations within a class is also examined.