Use this URL to cite or link to this record in EThOS:
Title: Analytically divergence-free discretization methods for Darcy's problem
Author: Schraeder, Daniela
ISNI:       0000 0004 2723 0930
Awarding Body: University of Sussex
Current Institution: University of Sussex
Date of Award: 2010
Availability of Full Text:
Access from EThOS:
Access from Institution:
Radial basis functions are well known for their applications in scattered data approximation and interpolation. They can also be applied in collocation methods to solve partial differential equations. We develop and analyse a mesh-free discretization method for Darcy's problem. Our approximation scheme is based upon optimal recovery, which leads to a collocation scheme using divergence-free positive denite kernels. Besides producing analytically incompressible flow fields, our method can be of arbitrary order, works in arbitrary space dimension and for arbitrary geometries. Firstly we establish Darcy's problem. To introduce the scheme we review and study divergence-free and curl-free matrix-valued kernels and their reproducing kernel Hilbert spaces. After developing the scheme, we find the approximation error for smooth target functions and the optimal approximation orders. Furthermore, we develop Sobolev-type error estimates for target functions rougher than the approximating function and show that the approximation properties extend to those functions. To find these error estimates, we apply band-limited approximation. Finally, we illustrate the method with numerical examples.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: QA Mathematics