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On Iterative Solution of Elliptic Difference Equations on a Mesh-Connected Array of Processors

Elementary operations such as inner products and vector norms used in conjugate gradient methods and for iteration error checking have an amazingly high complexity (computer time) on mesh connected arrays of processors because they require global information. Therefore iterative methods which require only local, nearest neighbor communications can be more efficient. The simplest such method, the Jacobi iterative method converges too slowly but we show how faster convergent methods based on Chebyshev iterative methods, with or without preconditioning can be applied efficiently for elliptic difference equations The preconditionings provide the bounds for the extreme eigenvalues, which are needed for the Chebyshev method and the number of iterations can be determined a priori, so no error checking is needed.