Eindige-element-metodes vir tydafhanklike parsiele differensiaalvergelykings

Abstract

This thesis is concerned with the numerical solutions of time-dependent differential equations by finite element methods. The solutions are approximated by finite elements which depend on both space and time variables. A new Galerkin method is formulated in which the deviations of the approximate solution in both space and time are simultaneously minimized in some sense. The investigation of the new method is mainly of a numerical and experimental nature. Where the analytical solutions of the problems considered are available, they are compared to the numerical solutions. If such solutions are not available then the numerical results will be compared to the solutions obtained by other known numerical methods. Firstly, a survey of finite element methods which are presently used to solve boundary value problems is presented. The Galerkin method and variations thereof will be emphasized. Next, we consider the application of finite element methods to onedimentional problems by using basis functions which are naturally dependent on time only. A step-by-step method is developed which forms the basis for a step-by-step method for two-dimensional problems. A generalisation of the one-step method leads to a Galerkin method in which the basic functions are dependent on both space and time variables. The method is applied to: (i) The heat equation with Dirichlet boundary conditions (ii) The convection-diffusion equation with periodic, Dirichlet and Neumann boundary conditions. In conclusion we apply this method to~ (i) The wave equation with Dirichlet boundary conditions and various different initial conditions. (ii) The wave equation with coupled boundary conditions.