Analysis of Iterative Methods for the Steady and Unsteady Stokes Problem: Application to Spectral Element Discretizations

A new and detailed analysis of the basic Uzawa algorithm for decoupling of the pressure and the velocity in the steady and unsteady Stokes operator is presented. The paper focuses on the following new aspects: explicit construction of the Uzawa pressure-operator spectrum for a semiperiodic model problem; general relationship of the convergence rate of the Uzawa procedure to classical inf-sup discretization analysis; and application of the method to high-order variational discretization

Development of symbolic algorithms for certain algebraic processes

This study investigates the problem of computing the exact greatest common divisor of two polynomials relative to an orthogonal basis, defined over the rational number field. The main objective of the study is to design and implement an effective and efficient symbolic algorithm for the general class of dense polynomials, given the rational number defining terms of their basis. From a general algorithm using the comrade matrix approach, the nonmodular and modular techniques are prescribed. If the coefficients of the generalized polynomials are multiprecision integers, multiprecision arithmetic will be required in the construction of the comrade matrix and the corresponding systems coefficient matrix. In addition, the application of the nonmodular elimination technique on this coefficient matrix extensively applies multiprecision rational number operations. The modular technique is employed to minimize the complexity involved in such computations. A divisor test algorithm that enables the detection of an unlucky reduction is a crucial device for an effective implementation of the modular technique. With the bound of the true solution not known a priori, the test is devised and carefully incorporated into the modular algorithm. The results illustrate that the modular algorithm illustrate its best performance for the class of relatively prime polynomials. The empirical computing time results show that the modular algorithm is markedly superior to the nonmodular algorithms in the case of sufficiently dense Legendre basis polynomials with a small GCD solution. In the case of dense Legendre basis polynomials with a big GCD solution, the modular algorithm is significantly superior to the nonmodular algorithms in higher degree polynomials. For more definitive conclusions, the computing time functions of the algorithms that are presented in this report have been worked out. Further investigations have also been suggested

Quadratic Curvature Gravity with Second Order Trace and Massive Gravity Models in Three Dimensions

The quadratic curvature lagrangians having metric field equations with second order trace are constructed relative to an orthonormal coframe. In $n>4$ dimensions, pure quadratic curvature lagrangian having second order trace constructed contains three free parameters in the most general case. The fourth order field equations of some of these models, in arbitrary dimensions, are cast in a particular form using the Schouten tensor. As a consequence, the field equations for the New massive gravity theory are related to those of the Topologically massive gravity. In particular, the conditions under which the latter is "square root" of the former are presented.Comment: 24 pages, to appear in GR

Techniques to accelerate boundary element contributions in elasticity

The problem of rapid re-analysis of small problems in elasticity is investigated. The aim is to enable updated stress contours to be displayed in real-time as a design geometry is dynamically modified. The focus of this work is small to medium sized problems; as a result it cannot be assumed that the solution phase dominates, and so the evaluation of boundary integrals is considered as well as the equation solution. Two strategies are employed for acceleration of boundary element integrals: the use of Look-Up Tables (LUTs) containing precomputed integrals and the use of approximate analytical expressions derived from surface fits. These may be used in the matrix assembly and internal point calculations. LUTs are derived for both flat and circular arc elements for both the displacement and stress boundary integral equation. Details are provided on suitable LUT refinements and the approach is benchmarked against conventional Gauss-Legendre quadrature. The surface fit approach is presented as an alternative to LUTs that does not incur the considerable memory cost associated with LUTs. This approach has been limited to flat elements. The equation solution is cast in a re-solution framework, in which we use a GM-RES iterative solver. Convergence is greatly accelerated by using an approximate but complete LU preconditioner updated periodically using multi-threading. Consideration of the period of update is investigated with reference to the spread of eigenvalues in the preconditioned system. The resulting system achieves the aim of providing real time update of contours for small to medium size problems on a PC. This development is expected to allow a qualitative change in the way engineers might use computer aided engineering tools, in which design ideas may rapidly be assessed immediately as a change is made