Maxim Larin

• PhD
• RWTH Aachen University

We consider a generalized Stokes equation with problem parameters ξ⩾0 (size of the reaction term) and ν>0 (size of the diffusion term). We apply a standard finite element method for discretization. The main topic of the paper is a study of efficient iterative solvers for the resulting discrete saddle point problem. We investigate a coupled multigri...

Today, multigrids and multilevel methods for solving a sparse linear system of equations are well known. They are both robust and efficient. O. Axelsson and M. Larin [J. Comput. Appl. Math. 89, No. 1, 135–153 (1998; Zbl 0941.65036)] have proposed the algebraic multilevel iteration (AMLI) method for finite element matrices. However, this method has...

The iterative methods for partial algebraic symmetric eigenvalue problems are considered for sparse positive definite matrices which arise in approximation of 2D and 3D boundary value problems. The approach is based on subspace iterations, Rayleigh-Ritz method, and the variable-step preconditioned conjugate gradient algorithm, including algebraic m...

In the present paper an improved version of the algebraic multilevel iteration (AMLI) method for finite element matrices, which has been suggested in (6), is proposed. To improve the quality of the AMLI-preconditioner or, the same, speed up the rate of con- vergence the family of iterative parameters defined on an error compensation principle is su...

Recently the direct application of a multigrid technique for computing the smallest eigenvalue and its corresponding eigenvector of a large symmetric positive definite matrix A has been investigated in (5). This method solves the eigenvalue problems on the sequence of nested grids using an interpolant of the solution on each grid as the initial gue...

The present paper is devoted to an improvement of the theory of the recently proposed generalized augmented matrix preconditioning method (5). Namely, we compute a sharp lower bound on eigenvalues of the preconditioned matrix based on the properties of the projector involved in its definition.

An element-by-element implementation of a recently proposed variable-step multilevel preconditioning method for solving second-order elliptic boundary value problems is considered. A special technique based on the internal properties of the preconditioning are used for the analysis of the rate of convergence. Performance results on standard test pr...

In the early eighties the direct application of a multigrid technique for solving the partial eigenvalue problem of computing few of the smallest eigenvalues and their corresponding eigenvectors of a differential operator was proposed by A. Brandt, S. McCormick and J. Ruge [SIAM J. Sci. Stat. Comput. 4, 244–260 (1983; Zbl 0517.65083)]. In the prese...

An algorithm and a code are described for computing several eigenvalues and the corresponding eigenvectors of a large sparse symmetric positive definite (SPD) matrix which arises as a result of grid approximation of multi-dimensional boundary value problems. The preconditioned inverse iteration method is implemented by using the explicit incomplete...

Recently an algebraic multilevel incomplete factorization (AMLIF) method for solving large linear systems with Stieltjes matrices has been proposed. This method is a combination of two well-known techniques: algebraic multilevel (AMLI) and relaxed incomplete factorization. However, the efficiency of AMLIF method strongly depends on the choice of th...

To solve a sparse linear system of equations resulting from the finite element approximation of elliptic self-adjoint second-order boundary-value problems an algebraic multilevel iteration method is presented. The new method can be considered as an extension of methods, which have been defined by Axelsson and Eijkhout (1991) for nine-point matrices...

A new way of carrying out Cholesky factorization to solve five-point systems of equations on a rectangular grid, and the corresponding data structure, similar to the idea of nested bisections, is proposed. Dirichlet's problem for Poisson's equation in a rectangle is used as a model. A comparison is made with sparse row format and compact row storag...

The efficiency of incomplete factorization with conjugate-gradients acceleration is studied for difference Helmholtz’ equations on a sphere with a special numeration of the grid nodes. Economical implementations of the methods of cyclic and block-cyclic reduction are given. The converge rate of the method is estimated and numerical examples are giv...