Elias A. Lipitakis

A normalized implicit conjugate gradient method for the solution of large sparse systems of linear equations

Linear systems arising from the finite difference approximation of a general self-adjoint elliptic partial differential equation on the unit square and cube are considered. The regular and banded structure of the coefficient matrices allows an approximate factorization in which only the elements along the diagonal, the co-diagonal and the r outermost diagonals of the band are included. This approximate factorization is chosen as the basis to yield a normalized system to which the method of conjugate gradients is implicitly applied. Due to the fact that a good approximate inverse of the coefficient matrix is used, the convergence of the new method-the normalized implicit conjugate gradient (NICG) method is very much improved.

Solving Non-Linear Elliptic Difference Equations by Extendable Sparse Factorization Procedures

New extendable LU sparse factorization procedures are presented for the solution of non-linear elliptic difference equations. The derived iterative methods are shown to be both competitive and computationally efficient in comparison with existing schemes. Application of the methods on non-linear elliptic boundary value problems both in two and three space dimensions are discussed and numerical results are given.ZusammenfassungNeue, erweiterte Verfahren der schwachbesetzten LU-Faktorisierung für die Lösung nichtlinearer elliptischer Differenzengleichungen werden vorgestellt. Von den zugehörigen Iterationsverfahren wird gezeigt, daß sie im Vergleich mit bekannten Verfahren konkurrenzfähig und numerisch effizient sind. Die Anwendung der Methoden auf nichtlineare elliptische Randwertprobleme in zwei und drei Dimensionen wird diskutiert; numerische Ergebnisse werden angegeben.

The rate of convergence of an approximate matrix factorization semi-direct method

SummaryThe definition of acceleration parameters for the convergence of a sparseLU factorization semi-direct method is shown to be based on lower and upper bounds of the extreme eigevalues of the iteration matrix. Optimum values of these parameters are established when the eigenvalues of the iteration matrix are either real or complex. Estimates for the computational work required to reduce theL2 norm of the error by a specified factor ɛ are also given.

Solving linear finite element systems by normalized approximate matrix factorization semi-direct methods

Abstract New normalized Extended to the Limit sparse factorization procedures in algorithmic form are derived yielding direct and iterative methods for the solution of finite element or finite difference systems of irregular structure. The proposed factorization procedures are chosen as the basis to yield normalized systems on which the Conjugate Gradient and Chebychev methods are implicitly applied. The application of the derived normalized implicit semi-direct methods on a two-dimensional elliptic boundary-value problem is discussed and numerical results are given.

Normalized factorization procedures for the solution of self-adjoint Elliptic partial differential equations in three-space dimensions

New normalized factorization procedures are presented for the coefficient matrix derived from the finite difference discretization of a self-adjoint elliptic 3D-P.D.E. leading to improved iterative schemes of solution. The derived algorithms are shown to be both competitive and computationally efficient in comparison with the existing schemes. Experimental results for a non-linear 3D magneto-hydrodynamic problem are given.

A normalized sparse linear equations solver

Normalized factorization procedures for the solution of large sparse linear finite element systems have been recently introduced in 3]. In these procedures the large sparse symmetric coefficient matrix of irregular structure is factorized exactly to yield a normalized direct solution method. Additionally, approximate factorization procedures yield implicit iterative methods for the finite difference or finite element solution. The numerical implementation of these algorithms is presented here and FORTRAN subroutines for the efficient solution of the resulting large sparse symmetric linear systems of algebraic equations are given.

Implicit semi-direct methods based on root-free sparse factorization procedures

In this paper new extendable sparse symmetric factorisation procedures are presented for the solution of self adjoint elliptic partial differential equations. The derived iterative methods are shown to be both competitive and computationally efficient in comparison with existing schemes. The application of the methods to a linear and non-linear elliptic boundary value problem in 2 dimensions is discussed and numerical results given.

Solving elliptic boundary-value problems on parallel processors by approximate inverse matrix semi-direct methods based on the multiple explicit Jacobi iteration

Abstract Approximate inverse matrix semi-direct methods for solving numerically linear systems on parallel processors are presented. The derived first and second order iterative methods possessing a high level of parallelism are based on the multiple explicit Jacobi iteration and originated by the approximation of the economized Chebychev polynomial and Neumann series to the inverse matrix. The convergence analysis of the proposed stationary and non-stationary explicit iterative schemes is developed and numerical results for a model problem are given.

Normalized implicit methods for the solution of non-linear elliptic boundary value problems

Abstract New normalized implicit methods are presented for the solution of self-adjoint elliptic P.D.E.s in two space dimensions. These methods are used in inner—outer iterative procedures in conjunction with Picard and Newton methods leading to improved composite iterative schemes for the solution of nonlinear elliptic boundary value problems. Applications of the derived methods include a nonlinear 2D magnetohydrodynamic problem and the 2D-Troeschs problem.

The use of second degree normalized implicit conjugate gradient methods for solving large sparse systems of linear equations

Abstract Second degree normalized implicit conjugate gradient methods for the numerical solution of self-adjoint elliptic partial differential equations are developed. A proposal for the selection of certain values of the iteration parameters ϱi, γi involved in solving two and three-dimensional elliptic boundary-value problems leading to substantial savings in computational work is presented. Experimental results for model problems are given.