Finite element solutions to boundary value problems

Abstract

This thesis consists of two distinct parts which deal with two-point boundary value problems and parabolic problems, respectively. In Section 1 we examine the numerical solution of a two-point boundary value problem by a collocation method based on the consistency relationship of regular splines. An existence and convergence result is established which generalises the 0(h^2) convergence result of the cubic spline collocation scheme for the problem in question. Contrary to most previously documented finite element schemes this method employs splines that may be non-linear in structure. Consequently, by a judicious choice of regular spline, the dominant terms of the true solution may be imitated more accurately than by the conventional polynomial based splines. The scheme is implemented by an algorithm that examines the suitability of various classes of regular splines and determines the subsequent deployment of them. The second section investigates semi-discrete finite element schemes for approximating the linear parabolic equation. A standard finite element discretization is employed for the space variable whilst an A0-stable, linear multistep, multiderivative discretization scheme, (L.M.S.D.) is used in time. We consider both the homogeneous and the nonhomogeneous linear parabolic equations and derive optimal convergence results for the above schemes. The convergence results achieved with a k-step L.M.S.D. scheme, incorporating the first m derivatives, generalise and extend the studies of several authors who concentrate on the particular cases of linear multistep formulae, m-l, and one-step schemes, k=1. Ao-stable L.M.S.D. 's are constructed and their implementation procedures examined. The suitability of selecting a L.M.S.D. method, with m, k>1, in a semi-discrete Galerkin scheme is discussed, and its superiority over semi-discrete Galerkin schemes, that incorporate linear multistep or one-step formulae, is confirmed in several aspects. Finally, a class of quasi-linear parabolic equations is solved by a semi-discrete Galerkin scheme that is third order accurate in time. This method is based on a particular third order L.M.S.D. scheme and requires the solution of linearly algebraic systems of equations at each time level. Thus, we improve on all the previously documented linearised schemes as they are only second order accurate in time. All the schemes described in Section 2 are unconditionally stable.