An Application of Cyclic Reduction to Ritz Type Difference Equations

A. K. Rigler
1964 Mathematics of Computation
1. Introduction. Iterative methods are often preferred over direct methods for solving the large systems of linear algebraic equations which arise in the finite difference approximation of boundary value problems for elliptic partial differential equations. An important reason for this preference is that the nonzero elements of the coefficient matrix are quite sparse, occurring in narrow bands along and parallel to the main diagonal. Since an iterative method such as successive over-relaxation
more » ... ve over-relaxation leaves the coefficient matrix unchanged, it imposes a comparatively modest requirement on computer storage capacity. A procedure introduced by Schröder [4] reduces the number of unknowns and improves the convergence rate of relaxation methods applied to the reduced problem. Called "cyclic reduction" by Varga [5] and "decomposition" by Collatz [1], its essential feature is the transformation of the coefficient matrix to a block triangular form. Hageman [3] proves that a block Gauss-Seidel solution of the reduced system must converge in fewer steps than a block Gauss-Seidel solution of the original. The technique is quite fruitful in improving the efficiency of finite difference solutions of potential and diffusion type problems. However, in solving some problems, for example equations with mixed derivatives, the reduction may be impractical for the following reason. The sparseness of the original coefficient matrix does not necessarily imply that the reduced coefficient matrix will be sparse ; possibly the reduction would increase both the number of coefficients to be stored and the number] of arithmetic operations required to complete the solution. It is the purpose of this paper to show that equations derived for self-ad joint second order elliptic systems by using the Ritz method as described by Friedrichs [2] can be reduced without these adverse side effects.