Rate of convergence of the method of group over-relaxation for solving variational difference schemes for elliptic equations of order 2m in a two-dimensional region

Abstract

WE show that, to obtain the solution of the system to an accuracy O( h 2 m ) we need to perform O( ln h −1 h m ) iterations. The influence of errors in performing the arithmetic operations on the computational results is investigated. To preserve an accuracy O( h 2 m ) we show that the elementary arithmetic operation should have a relative accuracy of order h 7 m − 1.5 . We described in [1] a modified type of variational difference schemes for solving elliptic equations of order 2 m, m ⩾ 1, in an arbitrary region. Estimates of the accuracy and condition number were established (the condition number is of order h 2 m , where h is the mesh step). In the present paper we consider the solution of the schemes described in [1], based on Hermitian interpolations [2], for equations in a two-dimensional region, using the method of group over-relaxation. We estimate the number of iterations sufficient to achieve an accuracy O( h 2 m ) in the norm of space L 2, and we prove that this number is of order O( h − m ln h −1) for certain values of the relaxation parameter. To realize one iterative step, O( h −2) arithmetic operations are required. We also examine the numerical stability of the method, and show that, to obtain a solution with accuracy O( h 2 m ) in the norm of L 2, it is sufficient for the relative accuracy of the elementary arithmetic operations to be of order h 7 m − 1.5 . Aspects of the convergence of the method of over-relaxation were discussed in [3]; some of the devices used in [3] will be employed in the present paper.