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Numerical approximation of fractional powers of elliptic operators

Authors: Andrea Bonito and Joseph E. Pasciak
Journal: Math. Comp. 84 (2015), 2083-2110
MSC (2010): Primary 35S15, 65R20, 65N12, 65N50, 65N30
Published electronically: March 12, 2015
MathSciNet review: 3356020
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Abstract: We present and study a novel numerical algorithm to approximate the action of $ T^\beta :=L^{-\beta }$ where $ L$ is a symmetric and positive definite unbounded operator on a Hilbert space $ H_0$. The numerical method is based on a representation formula for $ T^{-\beta }$ in terms of Bochner integrals involving $ (I+t^2L)^{-1}$ for $ t\in (0,\infty )$.

To develop an approximation to $ T^\beta $, we introduce a finite element approximation $ L_h$ to $ L$ and base our approximation to $ T^\beta $ on $ T_h^\beta := L_h^{-\beta }$. The direct evaluation of $ T_h^{\beta }$ is extremely expensive as it involves expansion in the basis of eigenfunctions for $ L_h$. The above mentioned representation formula holds for $ T_h^{-\beta }$ and we propose three quadrature approximations denoted generically by $ Q_h^\beta $. The two results of this paper bound the errors in the $ H_0$ inner product of $ T^\beta -T_h^\beta \pi _h$ and $ T_h^\beta -Q_h^\beta $ where $ \pi _h$ is the $ H_0$ orthogonal projection into the finite element space. We note that the evaluation of $ Q_h^\beta $ involves application of $ (I+(t_i)^2L_h)^{-1}$ with $ t_i$ being either a quadrature point or its inverse. Efficient solution algorithms for these problems are available and the problems at different quadrature points can be straightforwardly solved in parallel. Numerical experiments illustrating the theoretical estimates are provided for both the quadrature error $ T_h^\beta -Q_h^\beta $ and the finite element error $ T^\beta -T_h^\beta \pi _h$.

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  • [1] C. Bacuta, J. H. Bramble, and J. E. Pasciak, New interpolation results and applications to finite element methods for elliptic boundary value problems, East-West J. Numer. Math. 9 (2001), no. 3, 179-198. MR 1862579 (2002h:41053)
  • [2] Oleg G. Bakunin, Turbulence and Diffusion, Springer Series in Synergetics, Springer-Verlag, Berlin, 2008. Scaling versus equations. MR 2450437 (2009j:76001)
  • [3] A. V. Balakrishnan, Fractional powers of closed operators and the semigroups generated by them, Pacific J. Math. 10 (1960), 419-437. MR 0115096 (22 #5899)
  • [4] Peter W. Bates, On some nonlocal evolution equations arising in materials science, Nonlinear dynamics and evolution equations, Fields Inst. Commun., vol. 48, Amer. Math. Soc., Providence, RI, 2006, pp. 13-52. MR 2223347 (2007g:35097)
  • [5] M. Sh. Birman and M. Z. Solomjak, Spectral Theory of Selfadjoint Operators in Hilbert Space, Mathematics and its Applications (Soviet Series), D. Reidel Publishing Co., Dordrecht, 1987. Translated from the 1980 Russian original by S. Khrushchëv and V. Peller. MR 1192782 (93g:47001)
  • [6] James H. Bramble and Xuejun Zhang, The analysis of multigrid methods, Handb. Numer. Anal., VII, North-Holland, Amsterdam, 2000, pp. 173-415. MR 1804746 (2001m:65183)
  • [7] Luis Caffarelli and Luis Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations 32 (2007), no. 7-9, 1245-1260. MR 2354493 (2009k:35096),
  • [8] P. Carr, H. Geman, D.B. Madan, and M. Yor, The fine structure of asset returns: An empirical investigation, Journal of Business, 75(2):305-332, APR 2002.
  • [9] Peter Constantin and Jiahong Wu, Behavior of solutions of 2D quasi-geostrophic equations, SIAM J. Math. Anal. 30 (1999), no. 5, 937-948. MR 1709781 (2000j:76019),
  • [10] Monique Dauge, Elliptic Boundary Value Problems on Corner Domains: Smoothness and Asymptotics of Solutions, Lecture Notes in Mathematics, vol. 1341, Springer-Verlag, Berlin, 1988. MR 961439 (91a:35078)
  • [11] G. Duvaut and J.-L. Lions, Inequalities in Mechanics and Physics, Springer-Verlag, Berlin, 1976. Translated from the French by C. W. John; Grundlehren der Mathematischen Wissenschaften, 219. MR 0521262 (58 #25191)
  • [12] A. Cemal Eringen, Nonlocal Continuum Field Theories, Springer-Verlag, New York, 2002. MR 1918950 (2003m:74003)
  • [13] Hiroshi Fujita and Takashi Suzuki, Evolution problems, Handb. Numer. Anal., II, North-Holland, Amsterdam, 1991, pp. 789-928. MR 1115241
  • [14] Ivan P. Gavrilyuk, An algorithmic representation of fractional powers of positive operators, Numer. Funct. Anal. Optim. 17 (1996), no. 3-4, 293-305. MR 1393162 (98b:47049),
  • [15] Ivan P. Gavrilyuk, Wolfgang Hackbusch, and Boris N. Khoromskij, $ \mathcal {H}$-matrix approximation for the operator exponential with applications, Numer. Math. 92 (2002), no. 1, 83-111. MR 1917366 (2003g:65061),
  • [16] Ivan P. Gavrilyuk, Wolfgang Hackbusch, and Boris N. Khoromskij, Data-sparse approximation to the operator-valued functions of elliptic operator, Math. Comp. 73 (2004), no. 247, 1297-1324. MR 2047088 (2005b:47086),
  • [17] Ivan P. Gavrilyuk, Wolfgang Hackbusch, and Boris N. Khoromskij, Data-sparse approximation to a class of operator-valued functions, Math. Comp. 74 (2005), no. 250, 681-708. MR 2114643 (2005i:65068),
  • [18] Ivan P. Gavrilyuk, Wolfgang Hackbusch, and Boris N. Khoromskij, Hierarchical tensor-product approximation to the inverse and related operators for high-dimensional elliptic problems, Computing 74 (2005), no. 2, 131-157. MR 2133692 (2006f:65049),
  • [19] Guy Gilboa and Stanley Osher, Nonlocal operators with applications to image processing, Multiscale Model. Simul. 7 (2008), no. 3, 1005-1028. MR 2480109 (2010b:94006),
  • [20] J.-L. Guermond, The LBB condition in fractional Sobolev spaces and applications, IMA J. Numer. Anal. 29 (2009), no. 3, 790-805. MR 2520170 (2010f:65256),
  • [21] M. Ilic, F. Liu, I. Turner, and V. Anh, Numerical approximation of a fractional-in-space diffusion equation. I, Fract. Calc. Appl. Anal. 8 (2005), no. 3, 323-341. MR 2252038 (2007e:35163)
  • [22] M. Ilic, F. Liu, I. Turner, and V. Anh, Numerical approximation of a fractional-in-space diffusion equation. II. With nonhomogeneous boundary conditions, Fract. Calc. Appl. Anal. 9 (2006), no. 4, 333-349. MR 2300467 (2007k:35535)
  • [23] R. B. Kellogg, Interpolation between subspaces of a Hilbert space, Technical report, Univ. of Maryland, Inst., Fluid Dynamics and App. Math., Tech. Note BN-719, 1971.
  • [24] John Lund and Kenneth L. Bowers, Sinc Methods for Quadrature and Differential Equations, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. MR 1171217 (93i:65004)
  • [25] B. M. McCay and M. N. L. Narasimhan, Theory of nonlocal electromagnetic fluids, Arch. Mech. (Arch. Mech. Stos.) 33 (1981), no. 3, 365-384 (English, with Russian and Polish summaries). MR 660727 (83f:76103)
  • [26] William McLean and Vidar Thomée, Iterative solution of shifted positive-definite linear systems arising in a numerical method for the heat equation based on Laplace transformation and quadrature, ANZIAM J. 53 (2011), no. 2, 134-155. MR 2966174,
  • [27] Sergey A. Nazarov and Boris A. Plamenevsky, Elliptic Problems in Domains with Piecewise Smooth Boundaries, de Gruyter Expositions in Mathematics, vol. 13, Walter de Gruyter & Co., Berlin, 1994. MR 1283387 (95h:35001)
  • [28] R. H. Nochetto, E. Otarola, and A. J. Salgado, A PDE approach to fractional diffusion in general domains: a priori error analysis. submitted.
  • [29] S. A. Silling, Reformulation of elasticity theory for discontinuities and long-range forces, J. Mech. Phys. Solids 48 (2000), no. 1, 175-209. MR 1727557 (2000i:74008),
  • [30] Vidar Thomée, Galerkin Finite Element Methods for Parabolic Problems, Springer Series in Computational Mathematics, vol. 25, Springer-Verlag, Berlin, 1997. MR 1479170 (98m:65007)
  • [31] Qianqian Yang, Ian Turner, Fawang Liu, and Milos Ilić, Novel numerical methods for solving the time-space fractional diffusion equation in two dimensions, SIAM J. Sci. Comput. 33 (2011), no. 3, 1159-1180. MR 2800568 (2012h:65169),
  • [32] Toshio Yoshida.
    Functional Analysis.
    Springer-Verlag, New York, 1995.

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Additional Information

Andrea Bonito
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368

Joseph E. Pasciak
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368

Received by editor(s): July 2, 2013
Received by editor(s) in revised form: September 4, 2013, and January 17, 2014
Published electronically: March 12, 2015
Article copyright: © Copyright 2015 American Mathematical Society

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