## Numerical approximation of fractional powers of elliptic operators

HTML articles powered by AMS MathViewer

- by Andrea Bonito and Joseph E. Pasciak;
- Math. Comp.
**84**(2015), 2083-2110 - DOI: https://doi.org/10.1090/S0025-5718-2015-02937-8
- Published electronically: March 12, 2015
- PDF | Request permission

## 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 $T^\beta -T_h^\beta \pi _h$.

## References

- 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** - Oleg G. Bakunin,
*Turbulence and diffusion*, Springer Series in Synergetics, Springer-Verlag, Berlin, 2008. Scaling versus equations. MR**2450437** - A. V. Balakrishnan,
*Fractional powers of closed operators and the semigroups generated by them*, Pacific J. Math.**10**(1960), 419–437. MR**115096** - 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** - 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** - James H. Bramble and Xuejun Zhang,
*The analysis of multigrid methods*, Handbook of numerical analysis, Vol. VII, Handb. Numer. Anal., VII, North-Holland, Amsterdam, 2000, pp. 173–415. MR**1804746** - 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**, DOI 10.1080/03605300600987306 - 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. - 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**, DOI 10.1137/S0036141098337333 - Monique Dauge,
*Elliptic boundary value problems on corner domains*, Lecture Notes in Mathematics, vol. 1341, Springer-Verlag, Berlin, 1988. Smoothness and asymptotics of solutions. MR**961439**, DOI 10.1007/BFb0086682 - G. Duvaut and J.-L. Lions,
*Inequalities in mechanics and physics*, Grundlehren der Mathematischen Wissenschaften, vol. 219, Springer-Verlag, Berlin-New York, 1976. Translated from the French by C. W. John. MR**521262** - A. Cemal Eringen,
*Nonlocal continuum field theories*, Springer-Verlag, New York, 2002. MR**1918950** - Hiroshi Fujita and Takashi Suzuki,
*Evolution problems*, Handbook of numerical analysis, Vol. II, Handb. Numer. Anal., II, North-Holland, Amsterdam, 1991, pp. 789–928. MR**1115241** - 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**, DOI 10.1080/01630569608816695 - Ivan P. Gavrilyuk, Wolfgang Hackbusch, and Boris N. Khoromskij,
*$\scr H$-matrix approximation for the operator exponential with applications*, Numer. Math.**92**(2002), no. 1, 83–111. MR**1917366**, DOI 10.1007/s002110100360 - 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**, DOI 10.1090/S0025-5718-03-01590-4 - 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**, DOI 10.1090/S0025-5718-04-01703-X - 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**, DOI 10.1007/s00607-004-0086-y - Guy Gilboa and Stanley Osher,
*Nonlocal operators with applications to image processing*, Multiscale Model. Simul.**7**(2008), no. 3, 1005–1028. MR**2480109**, DOI 10.1137/070698592 - J.-L. Guermond,
*The LBB condition in fractional Sobolev spaces and applications*, IMA J. Numer. Anal.**29**(2009), no. 3, 790–805. MR**2520170**, DOI 10.1093/imanum/drn028 - 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** - 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** - 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. - 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**, DOI 10.1137/1.9781611971637 - 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** - 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**, DOI 10.1017/S1446181112000107 - 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**, DOI 10.1515/9783110848915.525 - R. H. Nochetto, E. Otarola, and A. J. Salgado,
*A PDE approach to fractional diffusion in general domains*:*a priori error analysis*. submitted. - 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**, DOI 10.1016/S0022-5096(99)00029-0 - Vidar Thomée,
*Galerkin finite element methods for parabolic problems*, Springer Series in Computational Mathematics, vol. 25, Springer-Verlag, Berlin, 1997. MR**1479170**, DOI 10.1007/978-3-662-03359-3 - 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**, DOI 10.1137/100800634 - Toshio Yoshida.
*Functional Analysis*. Springer-Verlag, New York, 1995.

## Bibliographic Information

**Andrea Bonito**- Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368
- MR Author ID: 783728
- Email: bonito@math.tamu.edu
**Joseph E. Pasciak**- Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368
- Email: pasciak@math.tamu.edu
- 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
- © Copyright 2015 American Mathematical Society
- Journal: Math. Comp.
**84**(2015), 2083-2110 - MSC (2010): Primary 35S15, 65R20, 65N12, 65N50, 65N30
- DOI: https://doi.org/10.1090/S0025-5718-2015-02937-8
- MathSciNet review: 3356020