Adaptive Methods and Collocation by Splines for Solving Differential Equations

Abstract

Splines have been used to approximate the solutions of differential equations for a while. In the first part of this thesis, adaptive algorithms based on the finite element method and splines on triangulations with hanging vertices are introduced and tested. In the second part, spline-based collocation methods are investigated: ordinary collocation, a generalized collocation model in 1D and 2D, and least-squares collocation with splines on triangulations. In particular, existence, uniqueness and error bounds of the (generalized) collocation solutions in the cubic case are presented. An error bound for the least-squares collocation on triangulations in approximating the solutions of the Possion equation is also given. Numerical examples are provided in all of the mentioned cases.