COMPUTABLE ERROR BOUNDS AND THE RITZ METHOD IN ONE DIMENSION PLUS THE NUMERICAL APPROXIMATION OF A MODEL CONVECTION-DIFFUSION EQUATION
Abstract
The first part of this thesis is concerned with a posteriori error estimation for the numerical approximation, by the Ritz method, of the solutions of linear, self-adjoint, second-order, two-point boundary value problems, so-called linear source problems. The Ritz method is an orthogonal projection method on the energy space, which space, I show, is a reproducing kernel Hilbert space with kernel function the Green's function for the original boundary value problem. By interpreting the Ritz method as an instance of optimal recovery, I obtain bounds for the pointwise values, maximum norm, and least squares norm of the error that are the best possible given the information used in computing the approximation. These bounds contain some quantities that are not, in general, known, and to make practical use of the bounds, one must compute upper bounds for these quantities. I approach the problem of bounding these quantities from two different points of view: first, using complementary variational principles, and second, using the theory of kernel functions. Numerical results are given. The second part of this thesis is concerned with an efficient scheme for the numerical approximation of a linear, constant-coefficient, non-self-adjoint, second-order, elliptic partial differential equation, with Dirichlet boundary conditions, on a rectangle in two dimensions, a so-called convection-diffusion equation. When the convective, or first order, term of this equation is dominant, the numerical approximation is difficult. I derive a new five point discretization for this problem. This discretization follows in a natural way from a local integral representation of the true solution, and it has the property that its coefficients possess a strong upwind bias. Because of this property, the normally highly unstable but very efficient Marching Algorithm is here stable for the direct solution of the discrete problem when the mesh size is larger than the thickness of the boundary layer of the true solution.
Degree
Ph.D.
Subject Area
Mathematics
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