We conducted a six-month investigation of the design, analysis, and software implementation of a class of singularity-insensitive, scalable, parallel nonlinear iterative methods for the numerical solution of nonlinear partial differential equations. The solutions of nonlinear PDEs are often nonsmooth and have local singularities, such as sharp fronts. Traditional nonlinear iterative methods, such as Newton-like methods, are capable of reducing the global smooth nonlinearities at a nearly quadratic convergence rate but may become very slow once the local singularities appear somewhere in the computational domain. Even with global strategies such as line search or trust region the methods often stagnate at local minima of {parallel}F{parallel}, especially for problems with unbalanced nonlinearities, because the methods do not have built-in machinery to deal with the unbalanced nonlinearities. To find the same solution u* of F(u) = 0, we solve, instead, an equivalent nonlinearly preconditioned system G(F(u*)) = 0 whose nonlinearities are more balanced. In this project, we proposed and studied a nonlinear additive Schwarz based parallel nonlinear preconditioner and showed numerically that the new method converges well even for some difficult problems, such as high Reynolds number flows, when a traditional inexact Newton method fails.

The LINPACK Benchmark has been used as a yardstick in measuring the performance of the Top500 installed high-end computers. This benchmark was chosen because it is widely used and performance numbers are available for almost all relevant systems. The approach used in the LINPACK Benchmark is to solve a dense system of linear equations. For the Top500, the benchmark allows the user to scale the size of the problem and to optimize the software in order to achieve the best performance for a given machine. This performance does not reflect the overall performance of a given system, as no single number ever can. It does, however, reflect the performance of a dedicated system for solving a dense system of linear equations. Since the problem is very regular, the performance achieved is quite high, and the performance numbers give a good check of peak performance of a system. By measuring the actual performance for different problem sizes n, a user can get not only the maximal achieved performance R{sub max} for the problem size N{sub max} but also the problem size N{sub 1/2} where half of the performance R{sub max} is achieved. These numbers together with the theoretical peak performance R{sub …

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