Positivity-preserving multigrid and multilevel methods

Abstract

Multigrids methods are extremely effective algorithms for solving the linear systems that arise from discretization of many differential equations in computational mathematics. A multigrid method uses a hierarchy of representations of the problem to achieve its efficiency and fast convergence. Originally inspired by a problem in adaptive mesh generation, this thesis focuses on the application of multigrid methods to a range of problems where the solution is required to preserve some additional properties during the iteration process. The major contribution of this thesis is the development of multigrid methods with the additional feature of preserving solution positivity: We have formulated both a multiplicative form multigrid method and a modified unigrid algorithm with constraints that are able to preserve positivity of the approximate solution at every iteration while maintaining convergence properties typical of normal multigrid methods. We have applied these algorithms to the 1D adaptive mesh generation problem to guarantee mesh nonsingularity, to singularly perturbed semilinear reaction-diffusion equations to compute unstable solutions, and to nonlinear diffusion equations. Numerical results show that our algorithms are effective and also possess good convergence properties.