Construction of higher order accurate vortex and particle methods

The standard point vortex method has recently been shown to be of high order of accuracy for problems on the whole plane, when using a uniform initial subdivision for assigning the vorticity to the points. If obstacles are present in the flow, this high order deteriorates to first or second order. New vortex methods are introduced which are of arbitrary accuracy (under regularity assumptions) regardless of the presence of bodies and the uniformity of the initial subdivision

Analysis and convergence of the MAC scheme. Part 1: The linear problem

The MAC discretization of fluid flow is analyzed for the stationary Stokes equations. It is proved that the discrete approximations do in fact converge to the exact solutions of the flow equations. Estimates using mesh dependent norms analogous to the standard H(sup 1) and L(sup 2) norms are given for the velocity and pressure, respectively

Iterative methods for elliptic finite element equations on general meshes

Iterative methods for arbitrary mesh discretizations of elliptic partial differential equations are surveyed. The methods discussed are preconditioned conjugate gradients, algebraic multigrid, deflated conjugate gradients, an element-by-element techniques, and domain decomposition. Computational results are included

Theory and computation of electromagnetic transition matrix elements in the continuous spectrum of atoms

The present study examines the mathematical properties of the free-free ( f-f) matrix elements of the full electric field operator, of the multipolar Hamiltonian. Special methods are developed and applied for their computation, for the general case where the scattering wavefunctions are calculated numerically in the potential of the term-dependent (N-1) electron core, and are energy-normalized. It is found that, on the energy axis, the f-f matrix elements of the full operator have singularities of first order in the case of equal photoelectron energies. The numerical applications are for f-f transitions in Hydrogen and Neon, obeying electric dipole and quadrupole selection rules. In the limit of zero photon wave-number, the full operator reduces to the length form of the electric dipole approximation (EDA). It is found that the results for the EDA agree with those of the full operator, with the exception of a photon wave-number region about the singularity.Comment: 39 pages, 11 figure

On substructuring algorithms and solution techniques for the numerical approximation of partial differential equations

Substructuring methods are in common use in mechanics problems where typically the associated linear systems of algebraic equations are positive definite. Here these methods are extended to problems which lead to nonpositive definite, nonsymmetric matrices. The extension is based on an algorithm which carries out the block Gauss elimination procedure without the need for interchanges even when a pivot matrix is singular. Examples are provided wherein the method is used in connection with finite element solutions of the stationary Stokes equations and the Helmholtz equation, and dual methods for second-order elliptic equations

The Seventh Ecumenical Council and the veneration of icons in Orthodoxy

In the Orthodox Christian tradition, icons are not regarded as works of art; they are rather a visual gospel and windows into the spiritual realm. They are intended to assist believers to be more contemplative and prayerful. They guide believers into a life of prayer. There are, however, those who consider them to be idolatrous. Such a belief is erroneous, since the honouring of created beings does not detract from being totally devoted to the Creator in whose image they were created. Icons portraying God’s grace are sanctifying and help affirm the faith of Orthodox believers. Icons are a concrete theology that instructs and leads believers to a spiritual reality and ultimately sanctify them as they transform them. They ultimately serve as conduits to the healing of body and soul through the grace of God and are essentially a prelude to the final transfiguration of the world

New discretization and solution techniques for incompressible viscous flow problems

Several topics arising in the finite element solution of the incompressible Navier-Stokes equations are considered. Specifically, the question of choosing finite element velocity/pressure spaces is addressed, particularly from the viewpoint of achieving stable discretizations leading to convergent pressure approximations. The role of artificial viscosity in viscous flow calculations is studied, emphasizing work by several researchers for the anisotropic case. The last section treats the problem of solving the nonlinear systems of equations which arise from the discretization. Time marching methods and classical iterative techniques, as well as some modifications are mentioned