Keywords: parallel computing, mildly nonlinear systems, ILU factorizations, two-stage methods.

In this paper we consider the problem of solving mildly nonlinear systems where the matrix is nonsingular and the nonlinear function has a certain local smoothness properties. In order to generate efficient algorithms to solve these mildly nonlinear systems, in [1], nonlinear two-stage multisplitting methods were considered from a theoretical point of view. These methods are ideal for parallel processing and provide a very general setting to study parallel block methods including overlapping. In this paper we extend the work of [1] by considering the important and useful cases where incomplete LU (ILU) factorizations [2] are considered as a mean of constructing the inner splittings. In [1] the convergence of these methods was shown when the matrix A is either monotone or an H-matrix. More precisely, in the latter case H-compatible splittings were considered. In this paper, together with some preliminary results that will be needed, we present new convergence results of these methods for both monotone matrices and H-matrices. In particular we analyze the convergence when ILU factorizations [2] are used to obtain the inner splittings. Finally, we present some numerical experiments which illustrate the performance of these algorithms. We have analyzed the performance of these methods on both shared and distributed architectures. As our illustrative example we have considered a nonlinear elliptic partial differential equation, known as the Bratu problem [3]. In this problem, heat generation from a combustion process is balanced by heat transfer due to conduction. Among other questions, we have analyzed the behaviour of these methods in relation to the parallel computer systems used, the influence of the level of fill-in for the ILU factorizations on the execution time of these parallel nonlinear two-stage methods, and the influence of the number of inner iterations preformed on both the execution time and the number of iterations needed for convergence. The efficiencies obtained show a good degree of parallelism of the algorithms designed in this work.

1

Z.Z. Bai, V. Migallón, J. Penadés, D.B. Szyld, "Block and asynchronous two-stage methods for mildly nonlinear systems", Numerische Mathematik, 82, 1-20, 1999. doi:10.1007/s002110050409

2

J.A. Meijerink, A. van der Vorst, "An iterative solution method for linear systems of which the coefficient matrix is a symmetric M-matrix", Mathematics of Computation, 31(137), 148-162, 1977. doi:10.2307/2005786

3

B.M. Averick, R.G. Carter, J.J. More, G. Xue, "The MINPACK-2 Test Problem Collection", Technical Report MCS-P153-0692, Mathematics and Computer Science Division. Argonne National Laboratory, 1992.

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