A meshless Galerkin method for non-local diffusion using localized kernel bases

Lehoucq, R.; Narcowich, F.; Rowe, S.; Ward, J.

doi:10.1090/mcom/3294

A meshless Galerkin method for non-local diffusion using localized kernel bases
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by R. B. Lehoucq, F. J. Narcowich, S. T. Rowe and J. D. Ward;

Math. Comp. 87 (2018), 2233-2258

DOI: https://doi.org/10.1090/mcom/3294

Published electronically: February 6, 2018

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Abstract:

We introduce a meshless method for solving both continuous and discrete variational formulations of a volume constrained, non-local diffusion problem. We use the discrete solution to approximate the continuous solution. Our method is non-conforming and uses a localized Lagrange basis that is constructed out of radial basis functions. By verifying that certain inf-sup conditions hold, we demonstrate that both the continuous and discrete problems are well-posed, and also present numerical and theoretical results for the convergence behavior of the method. The stiffness matrix is assembled by a special quadrature routine unique to the localized basis. Combining the quadrature method with the localized basis produces a well-conditioned, symmetric matrix. This then is used to find the discretized solution.

References

Academy for Advanced Telecommunications and Learning Technologies, Texas A&M University Brazos HPC, http://brazos.tamu.edu, 2015.
Gabriel Acosta and Juan Pablo Borthagaray, A fractional Laplace equation: regularity of solutions and finite element approximations, SIAM J. Numer. Anal. 55 (2017), no. 2, 472–495. MR 3620141, DOI 10.1137/15M1033952
Robert A. Adams, Sobolev spaces, Pure and Applied Mathematics, Vol. 65, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR 450957
Burak Aksoylu and Tadele Mengesha, Results on nonlocal boundary value problems, Numer. Funct. Anal. Optim. 31 (2010), no. 12, 1301–1317. MR 2738853, DOI 10.1080/01630563.2010.519136
Ivo Babuška, Uday Banerjee, and John E. Osborn, Survey of meshless and generalized finite element methods: a unified approach, Acta Numer. 12 (2003), 1–125. MR 2249154, DOI 10.1017/S0962492902000090
Stephen D. Bond, Richard B. Lehoucq, and Stephen T. Rowe, A Galerkin radial basis function method for nonlocal diffusion, Meshfree methods for partial differential equations VII, Lect. Notes Comput. Sci. Eng., vol. 100, Springer, Cham, 2015, pp. 1–21. MR 3587375
Susanne C. Brenner and L. Ridgway Scott, The mathematical theory of finite element methods, 3rd ed., Texts in Applied Mathematics, vol. 15, Springer, New York, 2008. MR 2373954, DOI 10.1007/978-0-387-75934-0
N. Burch, M. D’Elia, and R.B. Lehoucq, The exit-time problem for a Markov jump process, The European Physical Journal Special Topics 223 (2014), no. 14, 3257–3271.
Marta D’Elia and Max Gunzburger, The fractional Laplacian operator on bounded domains as a special case of the nonlocal diffusion operator, Comput. Math. Appl. 66 (2013), no. 7, 1245–1260. MR 3096457, DOI 10.1016/j.camwa.2013.07.022
Qiang Du, Max Gunzburger, R. B. Lehoucq, and Kun Zhou, Analysis and approximation of nonlocal diffusion problems with volume constraints, SIAM Rev. 54 (2012), no. 4, 667–696. MR 3023366, DOI 10.1137/110833294
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
E. Fuselier, T. Hangelbroek, F. J. Narcowich, J. D. Ward, and G. B. Wright, Localized bases for kernel spaces on the unit sphere, SIAM J. Numer. Anal. 51 (2013), no. 5, 2538–2562. MR 3097032, DOI 10.1137/120876940
T. Hangelbroek, On local RBF approximation, Adv. Comput. Math. 37 (2012), no. 2, 285–299. MR 2944053, DOI 10.1007/s10444-011-9212-5
Thomas Hangelbroek, Francis J. Narcowich, Christian Rieger, and Joseph D. Ward, An inverse theorem for compact Lipschitz domains for using kernel bases, Math. Comp. (2016), In press.
Norbert Heuer and Thanh Tran, Radial basis functions for the solution of hypersingular operators on open surfaces, Comput. Math. Appl. 63 (2012), no. 11, 1504–1518. MR 2922049, DOI 10.1016/j.camwa.2012.03.038
R. B. Lehoucq and S. T. Rowe, A radial basis function Galerkin method for inhomogeneous nonlocal diffusion, Comput. Methods Appl. Mech. Engrg. 299 (2016), 366–380. MR 3434919, DOI 10.1016/j.cma.2015.10.021
Francis J. Narcowich, Stephen T. Rowe, and Joseph D. Ward, A novel Galerkin method for solving PDEs on the sphere using highly localized kernel bases, Math. Comp. 86 (2017), no. 303, 197–231. MR 3557798, DOI 10.1090/mcom/3097
Francis J. Narcowich, Joseph D. Ward, and Holger Wendland, Sobolev bounds on functions with scattered zeros, with applications to radial basis function surface fitting, Math. Comp. 74 (2005), no. 250, 743–763. MR 2114646, DOI 10.1090/S0025-5718-04-01708-9
Francis J. Narcowich, Joseph D. Ward, and Holger Wendland, Sobolev error estimates and a Bernstein inequality for scattered data interpolation via radial basis functions, Constr. Approx. 24 (2006), no. 2, 175–186. MR 2239119, DOI 10.1007/s00365-005-0624-7
Eleonora Di Nezza, Giampiero Palatucci, and Enrico Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math. 136 (2012), no. 5, 521–573. MR 2944369, DOI 10.1016/j.bulsci.2011.12.004
Robert Schaback, A unified theory of radial basis functions. Native Hilbert spaces for radial basis functions. II, J. Comput. Appl. Math. 121 (2000), no. 1-2, 165–177. Numerical analysis in the 20th century, Vol. I, Approximation theory. MR 1752527, DOI 10.1016/S0377-0427(00)00345-9
S. A. Silling, Linearized theory of peridynamic states, J. Elasticity 99 (2010), no. 1, 85–111. MR 2592410, DOI 10.1007/s10659-009-9234-0
Xiaochuan Tian and Qiang Du, Analysis and comparison of different approximations to nonlocal diffusion and linear peridynamic equations, SIAM J. Numer. Anal. 51 (2013), no. 6, 3458–3482. Author name corrected by publisher. MR 3143839, DOI 10.1137/13091631X
Holger Wendland, Scattered data approximation, Cambridge Monographs on Applied and Computational Mathematics, vol. 17, Cambridge University Press, Cambridge, 2005. MR 2131724