Biharmonic Equation Discretized Equations Pe

The block-five-diagonal system of algebraic equations Ax = f arise from the finite-difference approximation of the first biharmonic boundary value problem:The usual method of solving this kind of equations is the iterative method. In this paper, several new iterative methods will be given. They are to be discussed as follows:(1) The linear PE method is derived on the base of incompletely LU triangular decomposition. Based on the results, we obtain the linear PEkmethod by introducing the parameter k. Then the solvability and the convergence of the two methods are proved about the Hermite positive definite matrix and M-matrix. In the end, some examples are given to illustrate that the convergence of the two methods is better than some existing methods , such as Jacobi method and SBGS method. If the proper parameter is chosen, the linear PEk method is superior to the linear PE method.(2) In order to improve the linear PE method and the linear PEk method, the quadratic PE method and the quadratic PEk method are put forward. Then the solvability and the convergence of the two methods arediscussed about the Hermite positive definite matrix and M-matrix, and the interrelated conclusions are obtained. The numerical results in chapter four appear to show that the convergence of the quadratic PE method and the quadratic PE;method is better than the linear PE method and the linear PEk method.