The thesis studies the approximation properties of splines with maximum smoothness. We are interested in the behaviour of the approximation as the degree of the spline increases (so does its smoothness). By studying B-spline interpolation, we obtain error estimates measured in the semi-norm that are explicit in terms of mesh size, degree and smoothness. This new result also gives a higher approximation order than existing estimations. With the results, we investigate the B-spline finite element approximation with k-refinement, which is a strategy of improving the accuracy by increasing the degree and smoothness. The problem is studied in the setting of heat equations and wave equations. We give B-spline FEM schemes for the problems, and obtain error estimates. Moreover, by proving a Markov-type inequality for splines, where an exact constant is derived, we deduce how the stability of the scheme behaves with the k-refinements. We also improve the efficiency of the schemes for problems with periodic boundary conditions by applying the fast Fourier transform. The thesis also focuses on developing algorithms for efficiently evaluating the element system matrices in finite element methods with Berstein-Bâezier splines as shape functions, where the splines are of arbitrary order and defined on quadrilaterals and hexahedrons. The algorithms achieve the optimal complexity by making use of the sum factorial procedure. We test the algorithms in C++ implementation, and the numerical results illustrate that the optimal cost and expected accuracy are achieved.
|Date of Award||23 Feb 2016|
- University Of Strathclyde
|Supervisor||Desmond Higham (Supervisor) & John MacKenzie (Supervisor)|