Svetozar Margenov

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.

Algebraic multilevel preconditioning of anisotropic elliptic problems

In this paper the recently proposed algebraic multilevel iteration method for iterative solution of elliptic boundary value problems with anisotropy and discontinuous coefficients is studied. Based on a special approximation of the blocks corresponding to the new nodes at every discretization level, an optimal order preconditioner with respect to the arithmetic cost independent of both the discontinuity and the anisotropy of the coefficients is constructed. The advantages of the proposed algorithms are illustrated by numerical tests.

Uniform estimate of the constant in the strengthened CBS inequality for anisotropic non‐conforming FEM systems

Uniform estimate of the constant in the strengthened CBS inequality for anisotropic non-conforming FEM systems

Circulant block-factorization preconditioners for elliptic problems

New circulant block-factorization preconditioners are introduced and studied. The general approach is first formulated for the case of block tridiagonal sparse matrices. Then estimates of the relative condition number for a model Dirichlet boundary value problem are derived. In the case ofy-periodic problems the circulant block-factorization preconditioner is shown to give an optimal convergence rate. Finally, using a proper imbedding of the original Dirichlet boundary value problem to ay-periodic one a preconditioner of optimal convergence rate for the general case is obtained. The total computational cost of the preconditioner isO (N logN) (based on FFT), whereN is the number of unknowns. That is, the algorithm is nearly optimal. Various numerical tests that demonstrate the features of the circulant block-factorization preconditioners are presented.ZusammenfassungNeue zyklische Matrixzerlegungen werden eingeführt und untersucht. Der allgemeine Ansatz wird für den Fall blocktridiagonaler schwachbesetzter Matrizen formuliert. Danach werden Abschätzungen der relativen Konditionszahl für ein Dirichlet-Modellproblem abgeleitet. Es wird gezeigt, daß die zyklische Matrixzerlegung im Falley-periodischer Aufgaben optimale Konvergenzraten liefert. Nach Einbettung des ursprünglichen Dirichlet-Problems in einey-periodische Aufgabe erhält man den allgemeinen Fall. Der Gesamtaufwand des Präkonditionierers beträgtO (N logN) gemäß des FFT-Aufwandes, wobeiN die Zahl der Unbekannten ist. Damit ist der Algorithmus fast optimal. Verschiedene numerische Tests zeigen die Eigenschaften der zyklischen Matrixzerlegung.

Upper bound of the constant in the strengthened C.B.S. inequality for FEM 2D elasticity equations

The basic theory of the strengthened Cauchy–Buniakowskii–Schwarz (C.B.S.) inequality is the main tool in the convergence analysis of the recently proposed algebraic multilevel iterative methods. An upper bound of the constant γ in the strengthened C.B.S. inequality for the case of the finite element solution of 2D elasticity problems is obtained. It is assumed that linear triangle finite elements are used, the initial mesh consisting of right isosceles triangles and the mesh refinement procedure being uniform. For the resulting linear algebraic systems we have proved that γ2<0.75 uniformly on the mesh parameter and on Poissons ratio ν ϵ (0, 1/2). Furthermore, the presented numerical tests show that the same relation holds for arbitrary initial right triangulations, even in the case of degeneracy of triangles. The theoretical results obtained are practically important for successful implementation of the finite element method to large-scale modeling of complicated structures. They allow us to construct optimal order algebraic multilevel iterative solvers for a wide class of real–life elasticity problems.

Robust optimal multilevel preconditioners for non‐conforming finite element systems

We consider strategies to construct optimal order two- and multilevel hierarchical preconditioners for linear systems as arising from the finite element discretization of self-adjoint second order elliptic problems using non-conforming Crouzeix–Raviart linear elements. In this paper we utilize the hierarchical decompositions, derived in a previous work by the same authors (Numerical Linear Algebra with Applications 2004; 11:309–326) and provide a further analysis of these decompositions in order to assure robustness with respect to anisotropy. Finally, we show how to construct both multiplicative and additive versions of the algebraic multilevel iteration preconditioners and show robustness and optimal order convergence estimates. Copyright

Finite volume discretization of equations describing nonlinear diffusion in Li-Ion batteries

Numerical modeling of electrochemical process in Li-Ion battery is an emerging topic of great practical interest. In this work we present a Finite Volume discretization of electrochemical diffusive processes occurring during the operation of Li-Ion batteries. The system of equations is a nonlinear, time-dependent diffusive system, coupling the Li concentration and the electric potential. The system is formulated at length-scale at which two different types of domains are distinguished, one for the electrolyte and one for the active solid particles in the electrode. The domains can be of highly irregular shape, with electrolyte occupying the pore space of a porous electrode. The material parameters in each domain differ by several orders of magnitude and can be nonlinear functions of Li ions concentration and/or the electrical potential. Moreover, special interface conditions are imposed at the boundary separating the electrolyte from the active solid particles. The field variables are discontinuous across such an interface and the coupling is highly nonlinear, rendering direct iteration methods ineffective for such problems. We formulate a Newton iteration for a purely implicit Finite Volume discretization of the coupled system. A series of numerical examples are presented for different type of electrolyte/electrode configurations and material parameters. The convergence of the Newton method is characterized both as function of nonlinear material parameters and the nonlinearity in the interface conditions.

On the multilevel preconditioning of Crouzeix–Raviart elliptic problems

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

Robust Two-level Domain Decomposition Preconditioners for High-contrast Anisotropic Flows in Multiscale Media

Abstract In this paper we discuss robust two-level domain decomposition preconditioners for highly anisotropic heterogeneous multiscale problems. We present a construction of several coarse spaces that employ standard finite element and multiscale basis functions and discuss techniques to reduce the dimensions of coarse spaces without sacrificing the robustness. We experimentally study the performance of the preconditioner on a variety two-dimensional test problems with channels of high anisotropy. The numerical tests confirm the robustness of the perconditioner with respect to the underlying physical parameters.

Multilevel preconditioning of rotated bilinear non-conforming FEM problems

Preconditioners based on various multilevel extensions of two-level finite element methods (FEM) lead to iterative methods which often have an optimal order computational complexity with respect to the number of degrees of freedom of the system. Such methods were first presented in [O. Axelsson, P.S. Vassilevski, Algebraic multilevel preconditioning methods I, Numer. Math. 56 (1989) 157-177; O. Axelsson, P.S. Vassilevski, Algebraic multilevel preconditioning methods II, SIAM J. Numer. Anal. 27 (1990) 1569-1590] on (recursive) two-level splittings of the finite element space. The key role in the derivation of optimal convergence rate estimates is played by the constant @c in the so-called strengthened Cauchy-Bunyakowski-Schwarz (CBS) inequality, associated with the angle between the two subspaces of the splitting. More precisely, the value of the upper bound for @c@?(0,1) is a part of the construction of various multilevel extensions of the related two-level methods. In this paper algebraic two-level and multilevel preconditioning algorithms for second-order elliptic boundary value problems are constructed, where the discretization is done using Rannacher-Turek non-conforming rotated bilinear finite elements on quadrilaterals. An important point to make is that in this case the finite element spaces corresponding to two successive levels of mesh refinement are not nested in general. To handle this, a proper two-level basis is required to enable us to fit the general framework for the construction of two-level preconditioners for conforming finite elements and to generalize the method to the multilevel case. The proposed variants of the hierarchical two-level basis are first introduced in a rather general setting. Then, the parameters involved are studied and optimized. The major contribution of the paper is the derived estimates of the constant @c in the strengthened CBS inequality which is shown to allow the efficient multilevel extension of the related two-level preconditioners. Representative numerical tests well illustrate the optimal complexity of the resulting iterative solver.