Spline collocation on adaptive grids and non-rectangular domains.
Published: 2005
Language: eng
Number of pages: 187
Description: We first consider using optimal spline collocation on non-uniform partitions to solve one-dimensional linear second-order Boundary Value Problems (BVPs) that have solutions with regions of high variations, such as boundary layers. Optimal quadratic spline collocation (QSC) is extended to non-uniform partitions by using a mapping function from uniform to non-uniform partitions. Optimal cubic spline collocation (CSC) on non-uniform partitions is also examined. The formulation and analysis of the CSC methods are based, as in the QSC case, on a mapping function from uniform to non-uniform partitions. However, this method is converted into a mapping-free method and is integrated with adaptive techniques that include error and grid size estimators.Optimal spline collocation is a computationally cost effective finite element method that has been relatively recently developed. Although there are many important results for optimal spline collocation, we believe the method can further be extended to solve a series of more relevant and difficult problems.Spline collocation is then considered for solving two-dimensional linear second-order BVPs. We extend optimal spline collocation to solving BVPs on L- and T-shaped domains, which occur often in practice. The CSC method can handle general boundary conditions, which had previously never been done, but our approach requires a third step to achieve optimal convergence. We also consider two adaptive approaches for solving problems on rectangular domains whose solutions have rapid variations. As in the one-dimensional case, such problems present a significant challenge to obtain accurate approximations at minimal computational cost. We first implement a static method, which involves adding collocation points to regions where the solution has sharp variations, with CSC that uses skipped grids, which are grids that have several levels of refinement, in certain regions of the domain. An optimal skipped grid method is developed by converting the second-order BVP into a system of three first-order BVPs. We then implement a dynamic method, which involves moving the collocation points to regions where the solution has sharp variations, with CSC that incorporates the Moving Mesh Partial Differential Equation (MMPDE). The MMPDE method is further integrated into an adaptive method that can solve BVPs within a specific error tolerance.
Notes: Source: Dissertation Abstracts International, Volume: 66-10, Section: B, page: 5506.

Thesis (Ph.D.)--University of Toronto, 2005.

Electronic version licensed for access by U. of T. users.