1 Linear Equation Systems in the Numerical Solution of PDE’s 3 1.1 Examples of PDE’s . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Weak Formulation of Poisson’s Equation . . . . . . . . . . . . 6 1.3 Finite-Difference-Discretization of Poisson’s Equation . . . . . 7 1.4 FD Discretization for Convection-Diffusion . . . . . . . . . . 8 1.5 Irreducible and Diagonal Dominant Matrices . . . ...

Recasting this in the form above we have 1 B M N − = − and . 1 c M b − = It is easy to show that this iteration is consistent for any splitting as long as M is nonsingular. Obviously, to be practical the matrix M must be selected so that the system My d = is easily solved. Popular choices for M are diagonal matrices (as in the Jacobi method), lower triangular matrices (as in the Gauss-Seidel an...

We present some computational experimentation for solving a linear system Ax = b by applying the Triangle Algorithm described in [3], based on converting the system to a convex hull problem as described in [4]. A comparison of the performance with classical iterative methods such as Jacobi, Gauss-Seidel, and Successive Over-Relaxation method for small size problems suggests the Triangle Algorit...

In this paper, we present an analysis of a multigrid method for nonsym-metric and/or indeenite elliptic problems. In this multigrid method various types of smoothers may be used. One type of smoother which we consider is deened in terms of an associated symmetric problem and includes point and line, Jacobi and Gauss-Seidel iterations. We also study smoothers based entirely on the original opera...

This paper concerns the iterative solution of the linear system arising from the Chebyshev–collocation approximation of second-order elliptic equations and presents an optimal multigrid preconditioner based on alternating line Gauss–Seidel smoothers for the corresponding stiffness matrix of bilinear finite elements on the Chebyshev–Gauss–Lobatto grid.  2000 IMACS. Published by Elsevier Science...

Several stabilized finite element methods for the Stokes eigenvalue problem based on the lowest equal-order finite element pair are numerically investigated. They are penalty, regular, multiscale enrichment, and local Gauss integration method. Comparisons between them are carried out, which show that the local Gauss integration method has good stability, efficiency, and accuracy properties, and...

We develop a new method of implementing the Hartree-Fock calculations. A class of Gaussian bases is assumed, which includes the Kamimura-Gauss basis-set as well as the set equivalent to the harmonic-oscillator basis-set. By using the Fourier transformation to calculate the interaction matrix elements, we can treat various interactions in a unified manner, including finite-range ones. The presen...

a method for error estimation of isogeometric analysis of plane stress problems which is the based on stress recovery and using the super convergent properties of the gauss points, has been already introduced. this paper is devoted to the development of the method and study of the effects of these supercovergent points on the solution improvement and error estimation of axisymmetric problems by...

In this article, the properties of character sums, including Gauss sums and Jacobi sums are investigated.