Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • 1. S. M. Allen and J. W. Cahn, A macroscopic theory for antiphase boundary motion and its application to antiphase domain coarsing, Acta. Metal. 27 (1979), 1085-1095.
  • 2. T. Belytschko, Y. Krongauz, D. Organ, M. Fleming, and P. Krysl, Meshless methods: An overview and recent developments, Comput. Methods in Appl. Mechanics and Engineering 139 (1996), 3-47.
  • 3. H. Brezis and P. Mrionescu, Composition in fractional Sobolev spaces, Discrete and Continuous Dynamical Systems 7 (2001), 241-246. MR 1808397 (2002c:46063)
  • 4. G. Fasshauer, Meshfree approximation methods with MATLAB, World Scientific Publishers, Singapore, 2007. MR 2357267 (2008i:65002)
  • 5. N. Flyer and B. Fornberg, Radial basis functions: Developments and applications to planetary scale flows, Computers and Fluids 46 (2011), 23-32.
  • 6. N. Flyer and G. Wright, Transport schemes on a sphere using radial basis functions, J. Comp. Phys. 226 (2007), 1059-1084. MR 2356868 (2008h:86011)
  • 7. -, A radial basis function method for the shallow water equations on a sphere, Proc. R. Soc. A 465 (2009), 1949-1976. MR 2500804 (2010g:76093)
  • 8. B. Fornberg and E. Lehto, Stabilization of RBF-generated finite difference methods for convective PDEs, J. Comput. Phys. 230 (2011), 2270-2285. MR 2764546
  • 9. Q. T. Le Gia, Approximation of parabolic PDEs on spheres using collocation method, Adv. Comput. Math. 22 (2005), 377-397. MR 2131154 (2006a:65130)
  • 10. Q. T. Le Gia, F. J. Narcowich, J. D. Ward, and H. Wendland, Continuous and discrete least-squares approximation by radial basis functions on spheres, J. Approx. Theory 143 (2006), 124-133. MR 2271729 (2007k:41078)
  • 11. T. M. Morton and M. Neamtu, Error bounds for solving pseudodifferential equatons on spheres by collocation with zonal kernels, J. Approx. Theory 114 (2002), 242-268. MR 1883408 (2002k:65200)
  • 12. C. Müller, Spherical Harmonics, Springer, Berlin, 1966. MR 0199449 (33:7593)
  • 13. F. J. Narcowich, X. Sun, J. D. Ward, and H. Wendland, Direct and inverse Sobolev error estimates for scattered data interpolation via spherical basis functions, J. Foundations of Computational Mathematics 7 (2007), 369-390. MR 2335250 (2008i:41023)
  • 14. F. J. Narcowich and J. D. Ward, Scattered data interpolation on spheres: Error estimates and locally supported basis functions, SIAM J. Math. Anal. 33 (2002), 1393-1410. MR 1920637 (2003j:41021)
  • 15. J. E. Pearson, Complex patterns in a simple system, Science 261 (1993), 189-192.
  • 16. M. E. Taylor, Partial differential equations. III, Applied Mathematical Sciences, vol. 117, Springer, New York, 1997. MR 1477408 (98k:35001)
  • 17. V. Thomée, Galerkin finite element methods for parabolic problems, second ed., Springer, Berlin, 2006. MR 2249024 (2007b:65003)
  • 18. V. Thomée and L. Wahlbin, On Galerkin methods in semilinear parabolic problems, SIAM J. Numer. Anal. 12 (1975), 378-389. MR 0395269 (52:16066)
  • 19. H. Wendland, Scattered data approximation, Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, Cambridge, UK, 2005. MR 2131724 (2006i:41002)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2010): 35K58, 65M12, 65M15, 65M20

Retrieve articles in all journals with MSC (2010): 35K58, 65M12, 65M15, 65M20

Additional Information

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

American Mathematical Society