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

Wendland, Holger

doi:10.1090/S0025-5718-2012-02623-8

A high-order approximation method for semilinear parabolic equations on spheres
HTML articles powered by AMS MathViewer

by Holger Wendland PDF

Math. Comp. 82 (2013), 227-245 Request permission

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

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.
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.
Haim Brezis and Petru Mironescu, Composition in fractional Sobolev spaces, Discrete Contin. Dynam. Systems 7 (2001), no. 2, 241–246. MR 1808397, DOI 10.3934/dcds.2001.7.241
Gregory E. Fasshauer, Meshfree approximation methods with MATLAB, Interdisciplinary Mathematical Sciences, vol. 6, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2007. With 1 CD-ROM (Windows, Macintosh and UNIX). MR 2357267, DOI 10.1142/6437
N. Flyer and B. Fornberg, Radial basis functions: Developments and applications to planetary scale flows, Computers and Fluids 46 (2011), 23–32.
Natasha Flyer and Grady B. Wright, Transport schemes on a sphere using radial basis functions, J. Comput. Phys. 226 (2007), no. 1, 1059–1084. MR 2356868, DOI 10.1016/j.jcp.2007.05.009
Natasha Flyer and Grady B. Wright, A radial basis function method for the shallow water equations on a sphere, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 465 (2009), no. 2106, 1949–1976. MR 2500804, DOI 10.1098/rspa.2009.0033
Bengt Fornberg and Erik Lehto, Stabilization of RBF-generated finite difference methods for convective PDEs, J. Comput. Phys. 230 (2011), no. 6, 2270–2285. MR 2764546, DOI 10.1016/j.jcp.2010.12.014
Q. T. Le Gia, Approximation of parabolic PDEs on spheres using spherical basis functions, Adv. Comput. Math. 22 (2005), no. 4, 377–397. MR 2131154, DOI 10.1007/s10444-003-3960-9
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), no. 1, 124–133. MR 2271729, DOI 10.1016/j.jat.2006.03.007
Tanya M. Morton and Marian Neamtu, Error bounds for solving pseudodifferential equations on spheres by collocation with zonal kernels, J. Approx. Theory 114 (2002), no. 2, 242–268. MR 1883408, DOI 10.1006/jath.2001.3642
Claus Müller, Spherical harmonics, Lecture Notes in Mathematics, vol. 17, Springer-Verlag, Berlin-New York, 1966. MR 0199449
Francis J. Narcowich, Xingping Sun, Joseph D. Ward, and Holger Wendland, Direct and inverse Sobolev error estimates for scattered data interpolation via spherical basis functions, Found. Comput. Math. 7 (2007), no. 3, 369–390. MR 2335250, DOI 10.1007/s10208-005-0197-7
Francis J. Narcowich and Joseph D. Ward, Scattered data interpolation on spheres: error estimates and locally supported basis functions, SIAM J. Math. Anal. 33 (2002), no. 6, 1393–1410. MR 1920637, DOI 10.1137/S0036141001395054
J. E. Pearson, Complex patterns in a simple system, Science 261 (1993), 189–192.
Michael E. Taylor, Partial differential equations. III, Applied Mathematical Sciences, vol. 117, Springer-Verlag, New York, 1997. Nonlinear equations; Corrected reprint of the 1996 original. MR 1477408
Vidar Thomée, Galerkin finite element methods for parabolic problems, 2nd ed., Springer Series in Computational Mathematics, vol. 25, Springer-Verlag, Berlin, 2006. MR 2249024
Vidar Thomée and Lars Wahlbin, On Galerkin methods in semilinear parabolic problems, SIAM J. Numer. Anal. 12 (1975), 378–389. MR 395269, DOI 10.1137/0712030
Holger Wendland, Scattered data approximation, Cambridge Monographs on Applied and Computational Mathematics, vol. 17, Cambridge University Press, Cambridge, 2005. MR 2131724