| Abstract |
Algebraic multigrid (AMG) solvers of large sparse linear systems of
equations are based on the principles of multigrid but do not explicitly use the geometry of the grids. AMG may be regarded as the "next
generation" of multigrid: it is almost a "black box," requiring no specific tailoring for a new problem; plus, it can handle problems beyond
the scope of multigrid, for example, PDEs with disordered coefficients on highly
unstructured grids, and problems that do not have a PDE origin.
AMG's scope has been rather limited, though. Its coarsening procedures have been inadequate for general
nonscalar, or high-order, or nonelliptic PDE systems and also for
nonvariational discretizations. In this talk, we present a new algorithm for generating the coarse
levels in AMG, using the tool of "compatible relaxation." Given a certain system of equations, the
algorithm is capable of (a) assessing whether a coarse level in question is adequate for further use in
AMG and (b) constructing an adequate coarse level. We present the algorithm
and its applications for anisotropic, rotated-characteristic, and high-order elliptic equations. The algorithm can be incorporated into the
existing AMG codes. In addition, we discuss how to improve the other stages of AMG, which altogether might lead to a significant improvement
in linear solvers' performance.
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