Modified successive overrelaxation (MSOR) and equivalent 2-step iterative methods for collocation matrices

Abstract

We consider a class of consistently ordered matrices which arise from the discretization of Boundary Value Problems (BVPs) when the finite element collocation method, with Hermite elements, is used. Through a recently derived equivalence relationship for the asymptotic rates of convergence of the Modified Successive Overrelaxation (MSOR) and a certain 2-step iterative method, we determine the optimum values for the parameters of the MSOR method, as it pertains to collocation matrices. A geometrical algorithm, which utilizes 'capturing ellipse' arguments, has been successfully used. The fast convergence properties of the optimum MSOR method are revealed after its comparison to several well-known iterative schemes. Numerical examples, which include the solution of Poisson's equation, are used to verify our results.'"

Preconditioned modified AOR method for systems of linear equations

“…One may find some convergence results in [2][3][4][5] when A is a positive definite and strictly diagonally dominant matrix, for example, H -, L-, M-matrix, or a strictly or irreducible dominant matrix. Hadjidimos et al [6,7] generalized the MSOR method and proposed a class of modified accelerated over-relaxation (MAOR) methods when the matrix A is a generalized consistently ordered matrix. They also provided some convergence conditions for two-cyclic matrices.…”

Symmetric modified AOR method to solve systems of linear equations

“…Hadjidimos et al [8,9] generalized the MSOR method and proposed a class of modified accelerated overrelaxation (MAOR) methods when the matrix A is a generalized consistently ordered matrix and some convergence conditions for two-cyclic matrices were given. In [16], some sufficient and/or necessary conditions for convergence of the MAOR method have been established, when A is a Hermitian positive definite matrix, an H -, L-or M-matrix, and a strictly or irreducible diagonally dominant matrix.…”

Multilevel Preconditioners for Non--self-adjoint or Indefinite Orthogonal Spline Collocation Problems

“…, c, and f are sufficiently smooth, and a 12 = a 21 . The differential operator L satisfies the uniform ellipticity condition; that is, there is ν > 0 such that where L 2 (Ω) and H 2 (Ω) are the Sobolev spaces.…”

“…Classical iterative methods, such as Jacobi, Gauss-Seidel, and SOR, for the OSC solution of Poisson's equation on a uniform partition were studied in [21,26,37]. ADI methods for solving OSC problems with separable operators were investigated in [5,14,18].…”