Keywords: partial differential equation of elliptic type, Aitken acceleration of convergence, Schwarz domain decomposition, parallel computing, Darcy, Stokes.

This chapter is devoted to the numerical development of the Schwarz domain decomposition for non separable operators where the convergence is accelerated by the Aitken process which is not based on the mesh property. We demonstrate the linear divergence/convergence of the Schwarz method on the coupling of Darcy equation and Stokes vectorial equation. We derive explicit formula for the coefficient of amplification of the error. Then we recall the good property of this two-level domain decomposition especially in a metacomputing context. When the operator is non separable and/or the mesh is not regular, the technique for the acceleration cannot take advantage of a decomposition of the solution in "Fourier" or "sinus" basis leading to a diagonal or block diagonal acceleration. We propose three algorithms based on the singular value decomposition of the iterates produced by iterative methods exhibiting a pure linear convergence. Evidence of numerically efficient acceleration of convergence, based on a posteriori estimates provided by the singular value decomposition (SVD), are given for the Jacobi and Schwarz methods on Darcy problem where the permeability exhibits strong contrasts.

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