Stability of the Rayleigh-Ritz-Galerkin procedure for elliptic boundary value problems

Abstract

This thesis investigates the stability of the Rayleigh-Ritz-Galerkin procedure for the approximate solution of certain classes of linear and nonlinear elliptic boundary value problems. In numerical analysis literature over the last decade, piecewise Hermite and spline subspaces have often been proposed for the Rayleigh-Ritz-Galerkin procedure for the solution of elliptic boundary value problems . However, the use of pie cewise polynomial subspaces has not been investigated f rom the point of view of Mikhlin stability , and this thesis rectifies this neglect in the literature . In Chapter 2, we introduce the Rayleigh-Ritz, the Galerkin, the generalized Rayleigh-Ritz, and the generalized Galerkin methods for the approximate solution of linear operator equations. As well as the concept of Mikhlin stability for linear numerical processes, we also introduce Tucker stability for nonlinear numeric~l processes. Chapter 3 is concerned with certain classes of linear elliptic boundary value problems, where for each class, we establish basic stability theorems and then investigate the Mikhlin stability of the Rayleigh-Ritz-Galerkin procedure when the coordinate functions are appropriately scaled B-splines or elementary Hermites. The three classes that we consider are one dimensional, two dimensional, and multidimensional elliptic boundary value problems with Dirichlet boundary conditions. Chapter 4 is similar to Chapter 3 except that in this case, we are concerned with nonlinear elliptic boundary value problems. The first and second class considered are nonlinear two-point boundary value problems with Dirichlet and nonlinear boundary conditions, respectively. We also study a "model" nonlinear multidimensional problem. In Chapter 5, we study normalized uniformly asymptotically diagonal systems from the point of view of Mikhlin stability, and illustrate the type of instability that can arise with a numerical example . Show more