On an Alternating Direction Method for Solving the Plate Problem with Mixed Boundary Conditions

is of basic importance in the classical theory of plates. In [3] the authors consider an alternating direction method for solving the related biharmonic difference equation subject to the boundary conditions associated with a simply supported square plate. In.this case the deflection W and the second (normal) derivative IV,, are prescribed along the plate edge. Of practical importance for this method is the fact that estimates on the rate of convergence are obtained. :For the plate problem with mixed boundary conditions, machine results have indicated that in most cases the method converges equally well although the theory presented in [3] does not apply. In this note it will be sho~m that the method converges in a square region when, in addition to W being prescribed along the entire boundary, any combination of either Wn or Wn~ are prescribed, each along a complete side. Unfortunately, estimates on the rate of convergence are not~available to support the optimism raised by machine computations. However, for certain combinations of the boundary conditions, rather crude estimates show that the method converges at least half as fast as a similar problem for the s'/mply supported plate. The previous statements are based on a second order formulation of the boundary difference equations for the normal derivative. If first order approximations are used for W~, sharp estimates on the rate of convergence are still available. Finally, it should be remarked that, as in the case of Laplace's equation (see [1]), convergence estimates are not yet available for other than rectangular regions.