Petr Vanek

Algebraic Multigrid by Smoothed Aggregation for Second and Fourth Order Elliptic Problems

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.ZusammenfassungEs 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.

Convergence of Algebraic Multigrid Based on Smoothed Aggregation

Summary. We prove an abstract convergence estimate for the Algebraic Multigrid Method with prolongator defined by a disaggregation followed by a smoothing. The method input is the problem matrix and a matrix of the zero energy modes of the same problem but with natural boundary conditions. The construction is described in the case of a general elliptic system. The condition number bound increases only as a polynomial of the number of levels, and requires only a uniform weak approximation property for the aggregation operators. This property can be a-priori verified computationally once the aggregates are known. For illustration, it is also verified here for a uniformly elliptic diffusion equations discretized by linear conforming quasiuniform finite elements. Only very weak and natural assumptions on the hierarchy of aggregates are needed.

Energy Optimization of Algebraic Multigrid Bases

Abstract.We propose a fast iterative method to optimize coarse basis functions in algebraic multigrid by minimizing the sum of their energies, subject to the condition that linear combinations of the basis functions equal to given zero energy modes, and subject to restrictions on the supports of the coarse basis functions. For a particular selection of the supports, the first iteration gives exactly the same basis functions as our earlier method using smoothed aggregation. The convergence rate of the minimization algorithm is bounded independently of the mesh size under usual assumptions on finite elements. The construction is presented for scalar problems as well as for linear elasticity. Computational results on difficult industrial problems demonstrate that the use of energy minimal basis functions improves algebraic multigrid performance and yields a more robust multigrid algorithm than smoothed aggregation.

Two-grid Method for Linear Elasticity on Unstructured Meshes

We propose an abstract two-grid algorithm with convergence independent of the coarse-space size. The abstract algorithm is applied to problems of three-dimensional linear elasticity discretized on unstructured meshes. With no regularity assumptions we prove uniform convergence with respect to coarse-space size, domain, essential boundary conditions, and jumps in Young modulus. Numerical experiments confirm the theory and show that the method works well even if some assumptions of the theory are violated.

Detection of Strong Coupling in Algebraic Multigrid Solvers

Based on abstract convergence theory for the smoothed aggregation multigrid method [6], we present a new method for detecting strong connections (couplings) in matrices obtained by discretization (and subsequent coarsening) of elliptic problems. Although the coupling evaluation given here has been derived in connection with smoothed aggregation algorithm, it is fully applicable in any AMG method, providing zero-energy modes are available in the solver.

Multilevel Algebraic Elliptic Solvers

We survey some of the recent research in developing multilevel algebraic solvers for elliptic problems. A key concept is the design of a hierarchy of coarse spaces and related interpolation operators which together satisfy certain approximation and stability properties to ensure the rapid convergence of the resulting multigrid algorithms. We will discuss smoothed agglomeration methods, harmonic extension methods, and global energy minimization methods for the construction of these coarse spaces and interpolation operators.