Search

It is well-known that iterative methods of optimal order complexity with respect to the size of the system can be set up by utilizing preconditioners based on various multilevel extensions of two-level finite element methods (FEM), as was first shown in [5]. Thereby, the constant γ in the so-called Cauchy-Bunyakowski-Schwarz (CBS) inequality, which is associated with the angle between the two subspaces obtained from a (recursive) two-level splitting of the finite element space, plays a key role in the derivation of optimal convergence rate estimates. In this paper a generalization of an algebraic preconditioning algorithm for second-order elliptic boundary value problems is presented, where the domain is discretized using linear Crouzeix-Raviart finite elements and the two-level splitting is defined by differentiation and aggregation (DA). It is shown that the uniform estimate on the constant γ (as presented in [6]) can be improved if a minimum angle condition, which is an integral part in any mesh generator, is assumed to hold in the triangulation. The improved values of γ can then be exploited in the set up of more problem-adapted multilevel preconditioners with faster convergence rates. [ABSTRACT FROM AUTHOR]

Preconditioning techniques based on various multilevel extensions of two-level splittings of finite element (FE) spaces lead to iterative methods which have an optimal rate of convergence and computational complexity with respect to the number of degrees of freedom. This article deals with the construction of algebraic two-level and multilevel preconditioning algorithms for the Lamé equations of elasticity, which are discretized by Crouzeix-Raviart non-conforming linear finite elements on triangles. An important point to note is that in the non-conforming case the FE spaces corresponding to two successive levels of mesh refinements are not nested. To handle this, a proper aggregation-based two-level basis is considered, which enables us to fit the general framework of the two-level preconditioners and to generalize the method to the multilevel case. The derived estimate of the constant in the strengthened Cauchy-Bunyakowski-Schwarz (CBS) inequality is uniform with respect to both, mesh anisotropy and Poisson ratio, including the almost incompressible case. [ABSTRACT FROM AUTHOR]

The CGM is studied for nonsymmetric elliptic problems with mixed boundary conditions. The mesh independence of the convergence is in focus when symmetric part preconditioning is applied to the FEM discretizations of the BVP. Computations in 2 dimensions are presented to illustrate and complete the theoretical results. Keywords: Conjugate gradient method, preconditioning, symmetric part, mesh independence, mixed boundary conditions, experiments. [ABSTRACT FROM AUTHOR]

The main goal of this paper is to generalize the results, obtained in [1] to the case of quasi-linear ordinary differential equations of the second order with the third kind boundary conditions. It is a continuation of the paper series in which we have obtained weight a priori estimates of accuracy for difference schemes for linear parabolic type equations in one-dimensional [2] and two-dimensional [3] cases, quasi-linear parabolic type equations [4] and quasi-linear elliptic equations with conditions of the first kind [5]. In this paper it is shown that on approaching to the left or right boundary of the domain the rate of convergence of solution or it's first derivative, correspondingly, increases. The second accuracy order difference scheme of the special form has been used for this purpose. The paper is completed by numerical experiment, which results confirm theoretical statements. [ABSTRACT FROM AUTHOR]

In the present paper, we propose a hybrid parallel method for large sparse eigenvalue problems in a grid computing environment. A moment-based method that finds several eigenvalues and their corresponding eigenvectors in a given domain is used. This method is suitable for master-worker type parallel programming models. In order to improve the parallel efficiency of the method, we propose a hybrid implementation using a GridRPC system Ninf-G and MPI. We examined the performance of the proposed method in an environment where several PC clusters are used. [ABSTRACT FROM AUTHOR]

In this contribution we extend our previous results on the structured total least squares problem to the case of weighted cost functions. It is shown that the computational complexity of the proposed algorithm is preserved linear in the sample size when the weight matrix is banded with bandwidth that is independent of the sample size. [ABSTRACT FROM AUTHOR]