Error estimation of collocation solutions

Abstract

Partial differential equations (PDEs) often arise in mathematical models that explain, investigate, or predict real-world phenomena. The error-controlled numerical solution of time-dependent PDEs with one spatial dimension is an area that has seen much work, but the error-controlled numerical solution of PDEs with more than one spatial dimension has not had much focus. One of the goals of this thesis is to extend established algorithms for spatial error estimation of numerical solutions of PDEs in one dimension to PDEs in two dimensions. We focus on spatial error estimation schemes for numerical solutions obtained through the use of B-spline Gaussian collocation. This thesis also includes an investigation into the impact of error control on the computation of numerical solutions to a time-dependent, one-dimensional COVID-19 PDE model.