Abstract
Standard discretizations of Stokes problems lead to linear systems of equations in saddle point form, making difficult the application of algebraic multigrid methods. In this paper, a new approach is proposed. It consists in first transforming the system by pre- and post-multiplication with simple, algebraic, sparse block triangular matrices. This is a form of pre-conditioning in the literal sense, designed to make sure that the transformed matrix is well adapted to multigrid. In particular, after transformation, all the diagonal blocks are symmetric and positive definite, and correspond to, or resemble, a discrete Laplace operator. Then, to each of these diagonal blocks is associated a prolongation that works well for it, using any relevant algebraic or geometric multigrid method. Next, a multigrid scheme for the global system is naturally set up by combining these partial prolongations with a Galerkin coarse grid matrix. For this approach combined with damped Jacobi-smoothing, a uniform two-grid convergence bound is derived for the global system under the assumption that the two-grid schemes for the different diagonal blocks are themselves uniformly convergent. This result is illustrated by a few examples, showing further that time-dependent problems and variable viscosity can be handled in a natural way, without requiring parameter adjustment. A numerical comparison also shows that the new approach can be more effective than state-of-the-art block preconditioning techniques.
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Notes
If \(0<\alpha <2(\lambda _{\max }(D_A^{-1}A))^{-1}\), then \(2\alpha D_A^{-1}-\alpha ^2 D_A^{-1}A D_A^{-1}\) is SPD, and hence \(\widehat{C}\) is SPD as well when \(C\) is positive definite on the range of \(B^T\).
The offdiagonal entries of \(A_b^{(i)}\) correspond to the two offdiagonal entries of \(A\) that connect each node to other nodes belonging to the same aggregate (one on the same horizontal grid line and one on the same vertical grid line). In addition, each nodes has two connections that points outside the aggregate; altogether, these connections form the offdiagonal entries of \(A_r\).
Other algebraic multigrid methods often require less iterations while being overall slower because this result is achieved thanks to larger complexities; see [23] for a comparative discussion.
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Acknowledgments
I thank Artem Napov for useful comments and suggestions.
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Research Director of the Fonds de la Recherche Scientifique-FNRS.
