Abstract
An algebraic multigrid algorithm for symmetric, positive definite linear systems is developed based on the concept of prolongation by smoothed aggregation. Coarse levels are generated automatically. We present a set of requirements motivated heuristically by a convergence theory. The algorithm then attempts to satisfy the requirements. Input to the method are the coefficient matrix and zero energy modes, which are determined from nodal coordinates and knowledge of the differential equation. Efficiency of the resulting algorithm is demonstrated by computational results on real world problems from solid elasticity, plate bending, and shells.
Zusammenfassung
Es wird ein algebraisches Mehrgitterverfahren für symmetrische, positiv definite Systeme vorgestellt, das auf dem Konzept der geglätteten Aggregation beruht. Die Grobgittergleichungen werden automatisch erzeugt. Wir stellen eine Reihe von Bedingungen auf, die aufgrund der Konvergenztheorie heuristisch motiviert sind. Der Algorithmus versucht diese Bedingungen zu erfüllen. Eingabe der Methode sind die Matrix-Koeffizienten und die Starrkörperbewegungen, die aus den Knotenwerten unter Kenntnis der Differentialgleichung bestimmt werden. Die Effizienz des entstehenden Algorithmus wird anhand numerischer Resultate für praktische Aufgaben aus den Bereichen Elastizität, Platten und Schalen demonstriert.
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Vaněk, P., Mandel, J. & Brezina, M. Algebraic multigrid by smoothed aggregation for second and fourth order elliptic problems. Computing 56, 179–196 (1996). https://doi.org/10.1007/BF02238511
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DOI: https://doi.org/10.1007/BF02238511