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A high-order, analytically divergence-free discretization method for Darcy's problem


Authors: Daniela Schräder and Holger Wendland
Journal: Math. Comp. 80 (2011), 263-277
MSC (2010): Primary 65N15, 65N35
Published electronically: June 7, 2010
MathSciNet review: 2728979
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Abstract: We develop and analyze a meshfree discretization method for Darcy's problem. Our approximation scheme is based upon optimal recovery which leads to a collocation scheme using divergence-free positive definite kernels. Besides producing analytically incompressible flow fields, our method can be of arbitrary order, works in arbitrary space dimension and for arbitrary geometries. After deriving the scheme, we investigate the approximation error for smooth target functions and derive optimal approximation orders. Finally, we illustrate the method by giving numerical examples.


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Additional Information

Daniela Schräder
Affiliation: Department of Mathematics, University of Sussex, Brighton, BN1 9RF, England
Email: d.schraeder@sussex.ac.uk

Holger Wendland
Affiliation: Mathematical Institute, University of Oxford, 24-29 St Giles’, Oxford, OX1 3LB, England
Email: holger.wendland@maths.ox.ac.uk

DOI: https://doi.org/10.1090/S0025-5718-2010-02388-9
Received by editor(s): December 2, 2008
Received by editor(s) in revised form: November 2, 2009
Published electronically: June 7, 2010
Article copyright: © Copyright 2010 American Mathematical Society