Quadratic Spline Collocation Methods for Systems of Elliptic PDEs

Quadratic Spline Collocation Methods for Systems of Elliptic PDEs Kit Sun Ng Master of Science Graduate Department of Computer Science University of Toronto 2000 We consider Quadratic Spline Collocation (QSC) methods for solving systems of two linear second-order PDEs in two dimensions. Optimal order approximation to the solution is obtained, in the sense that the convergence order of the QSC approximation is the same as the order of the quadratic spline interpolant. We study the matrix properties of the linear system arising from the discretization of systems of two PDEs by QSC. We give su cient conditions under which the QSC linear system is uniquely solvable and the optimal order of convergence for the QSC approximation is guaranteed. We develop fast direct solvers based on Fast Fourier Transforms (FFTs) and iterative methods using multigrid or FFT preconditioners for solving the above linear system. Numerical results demonstrate that the QSC methods are fourth order locally on certain points and third order globally, and that the computational complexity of the linear solvers developed is almost asymptotically optimal. The QSC methods are compared to conventional second order discretization methods and are shown to produce smaller approximation errors in the same computation time, while they achieve the same accuracy in less time. ii Dedication To my family iii Acknowledgements First and foremost, I would like to thank my supervisor Professor C. Christara for her hard work and dedication in guiding me throughout this thesis. Without her help, this thesis would be far inferior. Secondly, I would like to thank my second reader Professor W. Enright for his helpful comments. Finally, I would like to thank the University of Toronto, OGSST, and Professor C. Christara for the nancial support. iv