Partitioning of subdomains in a finite element method

Abstract

This thesis consists of two main parts. Firstly, the ideas of the finite element method (FEM) are described with emphasis on the numerical principles incorporated implicitly in the method. Next, special attention is given to one of these methods - the collocation technique - which can be implemented simply in the practical framework of the FEM. We seek the "best" possible rate of convergence that can be obtained by a judicious selection of (i) the partition of the model domain and (ii) the choice of collocation points. Towards this objective, the collocation method is applied to two problems: (a) m th -order linear ordinary differential equation with homogeneous boundary conditions on the real line, and b) Dirichlet's problem on the plane. For the first problem (a), it is found that orthogonal collocation yields superior convergence in (ii); various optimality results in (i) are also obtained for the simple cases u' = f, u" = f. In the second problem (b) we attempt to use bilinear piecewise polynomials as trial functions. The resulting collocation scheme, using a finite-difference approximation, is however not possible to solve analytically.