Analysis of Monotone Numerical Schemes

• Main Author: Nosov, Vladimir
• Other Authors: Kirkland, Stephen (Mathematics) Lui, Shaun (Mathematics)
• Published: 2016
• Subjects: Mathematics
• Online Access: http://hdl.handle.net/1993/31790

In the study of partial differential equations (PDEs) one rarely finds an analytical solution. But a numerical solution can be found using different methods such as finite difference, finite element, etc. The main issue using such numerical methods is whether the numerical solution will converge to the “real" analytical solution and if so how fast will it converge as we shrink the discretization parameter. In the first part of this thesis discrete versions of well known inequalities from analysis are used in proving the convergence of certain numerical methods for the one dimensional Poisson equation with Dirichlet boundary conditions and with Neumann boundary conditions. A matrix is monotone if its inverse exists and is non-negative. In the second part of the thesis we will show that finite difference discretization of two PDEs result in monotone matrices. The monotonicity property will be used to demonstrate stability of certain methods for the Poisson and Biharmonic equations. Convergence of all schemes is also shown. === October 2016