The development of rational basis functions for the finite element method

Abstract

This thesis is concerned with the development of rational basis functions for the finite element method to obtain the numerical solution of time- dependent partial differential equations. The concept of a rational basis function is introduced and it is found to be biased in the upstream direction. This, together with the fact that a rational function possesses better approximation abilities than a polynomial, motivated the application o-f rational basis functions in the Galerkin method to solve convection- dominated phenomena. This method gives rise to a rational difference scheme that approximates steep gradients and discontinuities satisfactorily without a stringent mesh refinement. The method is extended to higher-order rational basis functions and continuous Ilermite rational basis functions with continuous first derivatives. The investigation of the new method is mainly of a numerical and experimental nature. If the analytical solutions of the problems considered are available they are compared to the numerical solutions If the solutions are not available, the numerical results are compared to the solutions obtained by existing numerical methods. The rational method is applied to: (i) a stiff ordinary differential equation, (ii) the convection-diffusion equation with Dirichlet, periodic and Neumann boundary conditions, (iii) the Korteweg-de Vries equation, and (iv) linear and nonlinear hyperbolic equations. Convergence, consistency and stability properties of the rational difference schemes are investigated.