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Title: Multilevel collocation with radial basis functions
Author: Farrell, Patricio
ISNI:       0000 0004 6060 4316
Awarding Body: University of Oxford
Current Institution: University of Oxford
Date of Award: 2014
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In this thesis, we analyse multilevel collocation methods involving compactly supported radial basis functions. We focus on linear second-order elliptic bound- ary value problems as well as Darcy's problem. While in the former case we use scalar-valued positive definite functions for constructing multilevel approximants, in the latter case we use matrix-valued functions that are automatically divergence-free. A similar result is presented for interpolating divergence-free vector fields. Even though it had been observed more than a decade ago that the stationary setting, i.e. when the support radii shrink as fast as the mesh norm, does not lead to convergence, it was up to now an open question how the support radii should depend on the mesh norm to ensure convergence. For each case above, we answer this question here thoroughly. Furthermore, we analyse and improve the stability of the linear systems. And lastly, we examine the case when the approximant does not lie in the same space as the solution to the PDE.
Supervisor: Wendland, Holger ; Gillow, Kathryn Sponsor: Oxford Centre for Collaborative Applied Mathematics
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: Radial basis functions ; Approximation Theory ; Radial Basis Functions ; Numerical Analysis ; Multilevel ; Collocation