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A high-order approximation method for semilinear parabolic equations on spheres

Author: Holger Wendland
Journal: Math. Comp. 82 (2013), 227-245
MSC (2010): Primary 35K58, 65M12, 65M15, 65M20
Published electronically: June 14, 2012
MathSciNet review: 2983023
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Abstract: We analyse a discretisation method for solving (systems of) semilinear parabolic equations on Euclidean spheres. The approximation method is based upon a discretisation in space using spherical basis functions and can hence be of arbitrary order in space, provided that the true solution is sufficiently smooth, which is, at least locally in time, true for semilinear parabolic problems. We rigorously prove stability and convergence of the semi-discrete approximation, if basis functions with a prescribed Sobolev regularity are employed. For discretisation in time, we give two examples and prove the expected convergence orders in space and time for the fully discretised system.

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Holger Wendland
Affiliation: Mathematical Institute, University of Oxford, 24-29 St Giles’, Oxford, OX1 3LB, United Kingdom

Received by editor(s): September 20, 2010
Received by editor(s) in revised form: August 15, 2011
Published electronically: June 14, 2012
Article copyright: © Copyright 2012 American Mathematical Society

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