Second Order Strongly Implicit Symmetric Factorization Methods for the Solution of Elliptic Difference Equations
Paul E. Saylor · SIAM Journal on Numerical Analysis · 1974
H. L. Stone has proposed an iterative method, here called a factorization method, to solve elliptic difference equations. Because of the complexity of the algorithm defining the iteration matrix, Stone described the procedure as strongly implicit. Analysis of its convergence properties is difficult. One reason is that the iteration matrix is nonsymmetric. This motivates the construction of symmetric factorization methods. In this paper the class of strongly implicit symmetric factorization methods is determined. From this, it is possible to determine the class of such methods that are second order. This is a property of the Stone method that seems fundamental to its success. Since the iteration matrix of any of the resulting methods is defined by an algorithm with undesirable numerical properties, it may be inferred that no numerically satisfactory second order strongly implicit symmetric factorization method exists.