Fast iterative solvers for boundary value problems on a local spherical region

Boundary value problems on local spherical regions arise naturally in geophysics and oceanography when scientists model a physical quantity on large scales. Meshless methods using radial basis functions provide a simple way to construct numerical solutions with high accuracy. However, the linear systems arising from these methods are usually ill-conditioned, which poses a challenge for iterative solvers. We construct preconditioners based on an additive Schwarz method to accelerate the solution process for solving boundary value problems on local spherical regions. References D. Crowdy. Point vortex motion on the surface of a sphere with impenetrable boundaries. Physics of Fluids, 18:036602 (2006). doi:10.1063/1.2183627. A. E. Gill. Atmosphere-Ocean Dynamics, International Geophysics Series Volume 30. Academic, New York (1982). R. Kidambi and P. K. Newton. Point vortex motion on a sphere with solid boundaries. Physics of Fluids, 12:581 (2000). doi:10.1063/1.870263. Q. T. Le Gia, I. H. Sloan, and T. Tran. Overlapping additive Schwarz preconditioners for elliptic PDEs on the unit sphere. Math. Comp., 78:79--101 (2009). doi:10.1090/S0025-5718-08-02150-9. C. Muller. Spherical Harmonics, Vol. 17 of Lecture Notes in Mathematics. Springer-Verlag, Berlin (1966). M. V. Nezlin. Some remarks on coherent structures out of chaos in planetary atmospheres and oceans. Chaos, 4:109--111 (1994). doi:10.1063/1.165997. T. Tran, Q. T. Le Gia, I. H. Sloan, and E. P. Stephan. Preconditioners for pseudodifferential equations on the sphere with radial basis functions. Numer. Math., 115:141--163 (2009). doi:10.1007/s00211-009-0269-8

Efficient solvers for the shallow water equations on a sphere

We present a finite element method using spherical splines to solve the shallow water equations on a sphere involving satellite data. We compare the proposed method with a meshless method using radial basis functions. The use of either radial basis functions or spherical splines leads to ill-conditioned systems of linear equations. To accelerate the solution process we use additive Schwarz and alternate triangular preconditioners. Some numerical experiments are presented to show the effectiveness of both preconditioners. References P. Alfeld, M. Neamtu and L. L. Schumaker. Bernstein--Bezier polynomials on spheres and sphere-like surfaces. Comput. Aided Geom. Design, 13:333--349, 1996. doi:10.1016/0167-8396(95)00030-5 P. Alfeld, M. Neamtu and L. L. Schumaker. Dimension and local bases of homogeneous spline spaces. SIAM J. Math. Anal., 27:1482--1501, 1996. doi:10.1137/S0036141094276275 P. Alfeld, M. Neamtu and L. L. Schumaker. Fitting scattered data on sphere-like surfaces using spherical splines. J. Comput. Appl. Math., 73:5--43, 1996. doi:10.1016/0377-0427(96)00034-9 V. Baramidze, M. J. Lai and C. K. Shumaker. Spherical splines for data interpolation and fitting. SIAM J. Sci. Comput., 28:241--259, 2006. doi:10.1137/040620722 N. Flyer and G. 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., 465:1949--1976, 2009. doi:10.1098/rspa.2009.0033 J. Galewsky, R. K. Scott and L. M. Polvani. An initial-value problem for testing numerical models of the global shallow-water equations. Tellus A, 56:429--440, 2004. doi:10.1111/j.1600-0870.2004.00071.x A. Gelb and J. P. Gleeson. Spectral viscosity for shallow water equations in spherical geometry. Mon. Wea. Rev., 129:2346--2360, 2001. doi:10.1175/1520-0493(2001)129<2346:SVFSWE>2.0.CO;2 A. N. Konovalov. To the theory of the alternating triangle iteration method. Siberian Math. J., 43:439-457, 2002. doi:10.1023/A:1015455317080 A. N. Konovalov. The steepest descent method with an adaptive alternating triangular preconditioner. Diff. Equat., 40:1018-1028, 2004. doi:10.1023/B:DIEQ.0000047032.23099.e3 A. N. Konovalov. Optimal adaptive preconditioners in static problems of the linear theory of elasticity. Diff. Equat., 45:1044-1052, 2009. doi:10.1134/S0012266109070118 M. J. Lai and L. L. Schumaker. Spline Functions on Triangulations. Number v. 13 in Encyclopedia of Mathematics and Its Applications. Cambridge University Press, 2007. Q. T. Le Gia, I. H. Sloan and T. Tran. Overlapping additive Schwarz preconditioners for elliptic PDEs on the unit sphere. Math. Comp., 78:79--101, 2009. http://www.ams.org/journals/mcom/2009-78-265/S0025-5718-08-02150-9/ R. D. Nair. Diffusion experiments with a global discontinuous galerkin shallow-water model. Mon. Wea. Rev., 137:3339--3350, 2009. doi:10.1175/2009MWR2843.1 T. D. Pham and T. Tran. A domain decomposition method for solving the hypersingular integral equation on the sphere with spherical splines. Numer. Math., 120:117--151, 2012. doi:10.1007/s00211-011-0404-1 T. D. Pham, T. Tran and A. Chernov. Pseudodifferential equations on the sphere with spherical splines. Math. Models Methods Appl. Sci., 21:1933--1959, 2011. doi:10.1142/S021820251100560X T. D. Pham, T. Tran and S. Crothers. An overlapping additive Schwarz preconditioner for the Laplace--Beltrami equation using spherical splines. Adv. Comput. Math., 37:93--121, 2012. doi:10.1007/s10444-011-9200-9 A. A. Samarskii and E. S. Nikolaev. Numerical methods for grid equations. Vol. II. Birkhauser Verlag, Basel, 1989. Iterative methods, Translated from the Russian and with a note by Stephen G. Nash. T. Tran, Q. T. Le Gia, I. H. Sloan and E. P. Stephan. Preconditioners for pseudodifferential equations on the sphere with radial basis functions. Numer. Math., 115:141--163, 2010. doi:10.1007/s00211-009-0269-8 D. L. Williamson, J. B. Drake and P. N. Swarztrauber. The Cartesian method for solving partial differential equations in spherical geometry. Dynamics of Atmospheres and Oceans, 27:679--706, 1997. doi:10.1016/S0377-0265(97)00038-