Keywords: asynchronous parallel algoritm, high performance computing, multisplitting methods, Krylov method, Newton method, discretized pseudo-linear problem, large scale systems, nonlinear boundary value problems.

The present study is related to the analysis and application of mixed multisplitting methods to solve pseudo - linear stationary problems. These problems are stationary either intrinsically or as the result of the discretization of time evolution problems by implicit or semi-implicit time marching schemes. The considered problems are defined as an affine application AU-F perturbed by an increasing diagonal operator . In the following A has the property of being a large dimensional M-matrix, F is a vector, U is the unknown vector. Note that this type of problem occurs when solving elliptic, parabolic or hyperbolic second order boundary-value problems. Note also that the M-matrix property is well verified after discretization by classical finite differences, finite volumes or finite elements provided that the angle condition is verified. In this case the problem will be solved by a specific method corresponding to a local linearization and to the implementation of the iterative Newton method. Thus, a large sparse linear system must be solved. This linear system is then associated with a fixed point problem and we intend to solve it by asynchronous parallel iterations. Taking into account the properties of the matrix A and the operator’s monotony properties , it is shown that fixed point applications are contractive with respect to an uniform weighted norm, which ensures on the one hand the existence and uniqueness of the solution of the algebraic system to be solved and on the other hand the convergence of asynchronous parallel iterations towards the solution of the problem. In addition, in order to unify the presentation and analysis of the algorithm behavior, we consider multisplitting methods that unify the presentation of sub-domain methods, either to model sub-domain methods without overlap, or to model sub-domain methods with overlap such as Schwarz’s alternating method. These multisplitting methods are then applied to solve the target problem, the convergence analysis being still carried out by contraction techniques with respect to a weighted uniform norm. From an implementation point of view, we consider a mixed algorithm constituted by a two stage iterative algorithm, where the outer iteration is the multisplitting method and the inner iteration is the Krylov based iterative method, typically the GMRES method. As applications, we consider a diffusion convection problem perturbed by an increasing diagonal operator, the problem being solved by a mixed Newton - multisplitting method.

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