Johannes Kraus

Algebraic multilevel preconditioning of finite element matrices using local Schur complements

We consider an algebraic multilevel preconditioning technique for SPD matrices arising from finite element discretization of elliptic PDEs. In particular, we address the case of non-M matrices. The method is based on element agglomeration and assumes access to the individual element matrices. The left upper block of the considered multiplicative two-level preconditioner is approximated using incomplete factorization techniques. The coarse-grid element matrices are simply Schur complements computed from local neighbourhood matrices, i.e. small collections of element matrices. Assembling these local Schur complements results in a global Schur complement approximation that can be analysed by regarding (local) macro elements. These components, when combined in the framework of an algebraic multilevel iteration, yield a robust and efficient linear solver. The presented numerical experiments include also the Lame differential equation for the displacements in the two-dimensional plane-stress elasticity problem. Copyright

Robust Algebraic Multilevel Methods and Algorithms

This book deals with algorithms for the solution of linear systems of algebraic equations with large-scale sparse matrices, with a focus on problems that are obtained after discretization of partial differential equations using finite element methods. Provides a systematic presentation of the recent advances in robust algebraic multilevel methods. Can be used for advanced courses on the topic.

Additive Schur Complement Approximation and Application to Multilevel Preconditioning

This paper introduces an algorithm for additive Schur complement approximation (ASCA), which can be applied in various iterative methods for solving systems of linear algebraic equations arising from finite element (FE) discretization of partial differential equations (PDE). It is shown how the ASCA can be used to set up a nonlinear algebraic multilevel iteration (AMLI) method. This requires the construction of a linear (multiplicative) two-level preconditioner at each level. The latter is computed in the course of a simultaneous exact two-by-two block factorization of local (stiffness) matrices associated with a covering of the entire domain by overlapping subdomains. Unlike in Schwarz type domain decomposition methods, this method does not require a global coarse problem but instead uses local coarse problems to provide global communication. A robust condition number bound is proved for a particular partitioning of the nodes of a uniform grid. The presented numerical tests demonstrate that the ASCA, whe...

A multilevel method for discontinuous Galerkin approximation of three‐dimensional anisotropic elliptic problems

We construct optimal order multilevel preconditioners for interior-penalty discontinuous Galerkin (DG) finite element discretizations of three-dimensional (3D) anisotropic elliptic boundary-value problems. In this paper, we extend the analysis of our approach, introduced earlier for 2D problems (SIAM J. Sci. Comput., accepted), to cover 3D problems. A specific assembling process is proposed, which allows us to characterize the hierarchical splitting locally. This is also the key for a local analysis of the angle between the resulting subspaces. Applying the corresponding two-level basis transformation recursively, a sequence of algebraic problems is generated. These discrete problems can be associated with coarse versions of DG approximations (of the solution to the original variational problem) on a hierarchy of geometrically nested meshes. A new bound for the constant γ in the strengthened Cauchy–Bunyakowski–Schwarz inequality is derived. The presented numerical results support the theoretical analysis and demonstrate the potential of this approach. Copyright

Multilevel Preconditioning of Two-dimensional Elliptic Problems Discretized by a Class of Discontinuous Galerkin Methods

We present optimal-order preconditioners for certain discontinuous Galerkin (DG) finite element discretizations of elliptic boundary value problems. We consider two variants of hierarchical splittings where the underlying finite element meshes are geometrically nested. A specific assembling process is proposed which facilitates the local analysis of the angle between the related subspaces. By applying the corresponding two-level basis transformation recursively, a sequence of algebraic problems is generated that can be associated with a hierarchy of coarse versions of DG approximations of the original problem. New bounds for the constant

On the multilevel preconditioning of Crouzeix–Raviart elliptic problems

\gamma

Algebraic Multigrid Based on Computational Molecules, 2: Linear Elasticity Problems

in the strengthened Cauchy-Bunyakowski-Schwarz inequality are derived. The presented numerical results support the theoretical analysis and demonstrate the potential of this approach.

Algebraic Multilevel Preconditioners for the Graph Laplacian Based on Matching in Graphs

We consider robust hierarchical splittings of finite element spaces related to non-conforming discretizations using Crouzeix–Raviart type elements. As is well known, this is the key to the construction of efficient two- and multilevel preconditioners. The main contribution of this paper is a theoretical and an experimental comparison of three such splittings. Our starting point is the standard method based on differences and aggregates (DA) as introduced in Blaheta et al. (Numer. Linear Algebra Appl. 2004; 11:309–326). On this basis we propose a more general (GDA) splitting, which can be viewed as the solution of a constraint optimization problem (based on certain symmetry assumptions). We further consider the locally optimal (ODA) splitting, which is shown to be equivalent to the first reduce (FR) method from Blaheta et al. (Numer. Linear Algebra Appl. 2004; 11:309–326). This means that both, the ODA and the FR splitting, generate the same subspaces, and thus the local constant in the strengthened Cauchy–Bunyakowski–Schwarz inequality is minimal for the FR (respectively ODA) splitting. Moreover, since the DA splitting corresponds to a particular choice in the parameter space of the GDA splitting, which itself is an element in the set of all splittings for which the ODA (or equivalently FR) splitting yields the optimum, we conclude that the chain of inequalities γ⩽γ⩽γ⩽3/4 holds independently of mesh and/or coefficient anisotropy. Apart from the theoretical considerations, the presented numerical results provide a basis for a comparison of these three approaches from a practical point of view. Copyright

Multilevel preconditioning of rotated bilinear non-conforming FEM problems

This paper deals with a new approach in algebraic multigrid (AMG) for self-adjoint and elliptic problems arising from finite-element discretization of linear elasticity problems. Generalizing our approach for scalar problems [J. K. Kraus and J. Schicho, Computing, 77 (2006), pp. 57-75], we propose an edge-matrix concept for point-block systems of linear algebraic equations. This gives a simple and reliable method for the evaluation of the strength of nodal dependence that can be applied to symmetric positive definite non-M matrices. In this paper the edge-matrix concept is developed for and applied to (two- as well as three-dimensional) linear elasticity problems. We consider approximate splittings of the problem-related element matrices into symmetric positive semidefinite edge matrices of rank one. The reproduction of edge matrices on coarse levels offers the opportunity to combine classical coarsening techniques with robust (energy-minimizing) interpolation schemes: The “computational molecules” involved in this process are assembled from edge matrices. This yields a flexible new variant of AMG that allows also for an efficient solution of problems with discontinuous coefficients, e.g., for composite materials.

Polynomial of Best Uniform Approximation to 1/x and Smoothing in Two-level Methods

This paper presents estimates of the convergence rate and complexity of an algebraic multilevel preconditioner based on piecewise constant coarse vector spaces applied to the graph Laplacian. A bound is derived on the energy norm of the projection operator onto any piecewise constant vector space, which results in an estimate of the two-level convergence rate where the coarse level graph is obtained by matching. The two-level convergence of the method is then used to establish the convergence of an algebraic multilevel iteration that uses the two-level scheme recursively. On structured grids, the method is proved to have convergence rate