Block iterative methods for the numerical solution of three dimensional mildly non-linear biharmonic problems of first kind

Abstract

In this article, we discuss two sets of new finite difference methods of order two and four using 19 and 27 grid points, respectively over a cubic domain for solving the three dimensional nonlinear elliptic biharmonic problems of first kind. For both the cases we use block iterative methods and a single computational cell. The numerical solution of (∂u/∂n) are obtained as by-product of the methods and we do not require fictitious points in order to approximate the boundary conditions. The resulting matrix system is solved by the block iterative method using a tri-diagonal solver. In numerical experiments the proposed methods are compared with the exact solutions both in singular and non-singular cases.