Jan Mandel

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.

Optimal convergence properties of the FETI domain decomposition method

Abstract The Finite Element Tearing and Interconnecting (FETI) method is a practical and efficient domain decomposition (DD) algorithm for the solution of self-adjoint elliptic partial differential equations. For large-scale structural problems discretized with shell and beam elements, this method was found to outperform popular iterative algorithms and direct solvers on both serial and parallel computers, and to compare favorably with leading DD methods. In this paper, we discuss some numerical properties of the FETI method that were not addressed before. In particular, we show that the mathematical treatment of the floating subdomains and the specific conjugate projected gradient algorithm that characterize the FETI method are equivalent to the construction and solution of a coarse problem that propagates the error globally, accelerates convergence, and ensures a performance that is independent of the number of subdomains. We also show that when the interface problem is optimally preconditioned and the mesh is partitioned into well structured subdomains with good aspect ratios, the performance of the FETI method is also independent of the mesh size. However, we also argue that the FETI and other leading DD methods for unstructured problems lose in practice these scalability properties when the mesh contains junctures with rotational degrees of freedom, or the decomposition is irregular and characterized by arbitrary subdomain aspect ratios. Finally, we report that for realistic problems, optimal preconditioners are not necessarily computationally efficient and can be outperformed by non-optimal ones.

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.

The finite volume element method for diffusion equations on general triangulations

This paper develops discretization error estimates for the finite volume element method on general triangulations of a polygonal domain in

Efficient preconditioning for the p -version finite element method in two dimensions

\mathcal{R}^2

Convergence of a balancing domain decomposition by constraints and energy minimization

using a special type of control volume. The theory applies to diffusion equations of the form \[ \begin{gathered} - \nabla (A\nabla u) = f\quad {\text{in }}\Omega , \hfill \\ u = 0\quad {\text{on }}\partial \Omega . \hfill \\ \end{gathered} \] Under fairly general conditions, the theory establishes

Balancing domain decomposition for problems with large jumps in coefficients

O(h)

The two-level FETI method for static and dynamic plate problems Part I: An optimal iterative solver for biharmonic systems

estimates of the error in a discrete

On the Convergence of a Dual-Primal Substructuring Method

\mathcal{H}^1

Energy optimization of algebraic multigrid bases

seminorm. Under an additional assumption concerning local uniformity of the triangulation, the estimate is improved to