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Parallel algebraic multilevel Schwarz preconditioners for a class of elliptic PDE systems

Alfio Borzý (alfio.borzi***at***mathematik.uni-wuerzburg.de)
Valentina De Simone (valentina.desimone***at***unina2.it)
Daniela di Serafino (daniela.diserafino***at***unina2.it)

Abstract: We present algebraic multilevel preconditioners for linear systems arising from the discretization of systems of coupled elliptic partial differential equations (PDEs). These preconditioners are based on modifications of Schwarz methods and of the smoothed aggregation technique, where the coarsening strategy and the restriction and prolongation operators are defined using a point-based approach with a primary matrix corresponding to a single PDE. The preconditioners are implemented in a parallel computing framework and are tested on two representative PDE systems. The results of the numerical experiments show the effectiveness and the scalability of the proposed methods; in particular, when applied to the optimality systems associated with elliptic PDE-constrained optimization problems, these preconditioners appear to be rather insensitive to the regularization parameter in the cost functional. A convergence theory for the twolevel case is presented.

Keywords: systems of elliptic PDEs, algebraic multilevel preconditioners, Schwarz methods, smoothed aggregation, parallel computing

Category 1: Nonlinear Optimization (Systems governed by Differential Equations Optimization )


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Entry Submitted: 12/30/2011
Entry Accepted: 12/30/2011
Entry Last Modified: 10/13/2013

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