Schraeder, Daniela (2010) Analytically divergence-free discretization methods for Darcy's problem. Doctoral thesis (DPhil), University of Sussex.
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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.
|Item Type:||Thesis (Doctoral)|
|Schools and Departments:||School of Mathematical and Physical Sciences > Mathematics|
|Subjects:||Q Science > QA Mathematics|
|Depositing User:||Library Cataloguing|
|Date Deposited:||13 May 2010|
|Last Modified:||04 Aug 2015 13:02|