Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Multilevel methods for nonuniformly elliptic operators and fractional diffusion

Authors: Long Chen, Ricardo H. Nochetto, Enrique Otárola and Abner J. Salgado
Journal: Math. Comp. 85 (2016), 2583-2607
MSC (2010): Primary 65N55, 65F10, 65N22, 65N30, 35S15, 65N12
Published electronically: March 3, 2016
MathSciNet review: 3522963
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We develop and analyze multilevel methods for nonuniformly elliptic operators whose ellipticity holds in a weighted Sobolev space with an $ A_2$-Muckenhoupt weight. Using the so-called Xu-Zikatanov (XZ) identity, we derive a nearly uniform convergence result under the assumption that the underlying mesh is quasi-uniform. As an application we also consider the so-called $ \alpha $-harmonic extension to localize fractional powers of elliptic operators. Motivated by the scheme proposed by the second, third and fourth authors, we present a multilevel method with line smoothers and obtain a nearly uniform convergence result on anisotropic meshes. Numerical experiments illustrate the performance of our method.

References [Enhancements On Off] (What's this?)

  • [1] Thomas Apel and Joachim Schöberl, Multigrid methods for anisotropic edge refinement, SIAM J. Numer. Anal. 40 (2002), no. 5, 1993-2006 (electronic). MR 1950630 (2003m:65225),
  • [2] Daniel Arroyo, Alexei Bespalov, and Norbert Heuer, On the finite element method for elliptic problems with degenerate and singular coefficients, Math. Comp. 76 (2007), no. 258, 509-537. MR 2291826 (2008e:65336),
  • [3] 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)
  • [4] Zakaria Belhachmi, Christine Bernardi, and Simone Deparis, Weighted Clément operator and application to the finite element discretization of the axisymmetric Stokes problem, Numer. Math. 105 (2006), no. 2, 217-247. MR 2262757 (2008c:65310),
  • [5] M. Š. Birman and M. Z. Solomjak, Spektralnaya teoriya samosopryazhennykh operatorov v gilbertovom prostranstve, Leningrad. Univ., Leningrad, 1980 (Russian). MR 609148 (82k:47001)
  • [6] Andrea Bonito and Joseph E. Pasciak, Numerical approximation of fractional powers of elliptic operators, Math. Comp. 84 (2015), no. 295, 2083-2110. MR 3356020,
  • [7] James H. Bramble and Joseph E. Pasciak, New convergence estimates for multigrid algorithms, Math. Comp. 49 (1987), no. 180, 311-329. MR 906174 (89b:65234),
  • [8] James H. Bramble, Joseph E. Pasciak, Jun Ping Wang, and Jinchao Xu, Convergence estimates for multigrid algorithms without regularity assumptions, Math. Comp. 57 (1991), no. 195, 23-45. MR 1079008 (91m:65158),
  • [9] James H. Bramble, Joseph E. Pasciak, Jun Ping Wang, and Jinchao Xu, Convergence estimates for product iterative methods with applications to domain decomposition, Math. Comp. 57 (1991), no. 195, 1-21. MR 1090464 (92d:65094),
  • [10] 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 (2001m:65183)
  • [11] James H. Bramble and Xuejun Zhang, Uniform convergence of the multigrid $ V$-cycle for an anisotropic problem, Math. Comp. 70 (2001), no. 234, 453-470. MR 1709148 (2001g:65134),
  • [12] Achi Brandt, Multi-level adaptive solutions to boundary-value problems, Math. Comp. 31 (1977), no. 138, 333-390. MR 0431719 (55 #4714)
  • [13] Achi Brandt, Multigrid Techniques: 1984 Guide with Applications to Fluid Dynamics, GMD-Studien [GMD Studies], vol. 85, Gesellschaft für Mathematik und Datenverarbeitung mbH, St. Augustin, 1984. MR 772748 (87c:65139b)
  • [14] 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 (2008m:65001)
  • [15] A. Bueno-Orovio, D. Kay, V. Grau, B. Rodriguez, and K. Burrage, Fractional diffusion models of cardiac electrical propagation: role of structural heterogeneity in dispersion of repolarization, J. R. Soc. Interface, 11(97), 2014.
  • [16] Xavier Cabré and Yannick Sire, Nonlinear equations for fractional Laplacians II: Existence, uniqueness, and qualitative properties of solutions, Trans. Amer. Math. Soc. 367 (2015), no. 2, 911-941. MR 3280032,
  • [17] 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),
  • [18] Antonio Capella, Juan Dávila, Louis Dupaigne, and Yannick Sire, Regularity of radial extremal solutions for some non-local semilinear equations, Comm. Partial Differential Equations 36 (2011), no. 8, 1353-1384. MR 2825595 (2012h:35361),
  • [19] P. Carr, H. Geman, D. B. Madan, and M. Yor, The fine structure of asset returns: An empirical investigation, Journal of Business, 75 (2002), 305-33.
  • [20] L. Chen, $ i$FEM: An integrated finite element methods package in matlab, Technical report, University of California at Irvine, 2009.
  • [21] Long Chen, Deriving the X-Z identity from auxiliary space method, Domain decomposition methods in science and engineering XIX, Lect. Notes Comput. Sci. Eng., vol. 78, Springer, Heidelberg, 2011, pp. 309-316. MR 2867674,
  • [22] W. Chen, A speculative study of $ 2/3$-order fractional laplacian modeling of turbulence: Some thoughts and conjectures, Chaos 16 (2006), no. 2, 1-11.
  • [23] Durkbin Cho, Jinchao Xu, and Ludmil Zikatanov, New estimates for the rate of convergence of the method of subspace corrections, Numer. Math. Theory Methods Appl. 1 (2008), no. 1, 44-56. MR 2401666 (2009b:65077)
  • [24] Philippe G. Ciarlet, The Finite Element Method for Elliptic Problems, Classics in Applied Mathematics, vol. 40, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002. Reprint of the 1978 original [North-Holland, Amsterdam; MR0520174 (58 #25001)]. MR 1930132
  • [25] J. Cushman and T. Glinn, Nonlocal dispersion in media with continuously evolving scales of heterogeneity, Trans. Porous Media 13 (1993), 123-138.
  • [26] Qiang Du, Max Gunzburger, R. B. Lehoucq, and Kun Zhou, Analysis of the volume-constrained peridynamic Navier equation of linear elasticity, J. Elasticity 113 (2013), no. 2, 193-217. MR 3102595,
  • [27] G. Duvaut and J.-L. Lions, Inequalities in Mechanics and Physics, translated from the French by C. W. John, Grundlehren der Mathematischen Wissenschaften, 219, Springer-Verlag, Berlin-New York, 1976. MR 0521262 (58 #25191)
  • [28] A. Cemal Eringen, Nonlocal Continuum Field Theories, Springer-Verlag, New York, 2002. MR 1918950 (2003m:74003)
  • [29] Alexandre Ern and Jean-Luc Guermond, Theory and Practice of Finite Elements, Applied Mathematical Sciences, vol. 159, Springer-Verlag, New York, 2004. MR 2050138 (2005d:65002)
  • [30] Eugene B. Fabes, Carlos E. Kenig, and Raul P. Serapioni, The local regularity of solutions of degenerate elliptic equations, Comm. Partial Differential Equations 7 (1982), no. 1, 77-116. MR 643158 (84i:35070),
  • [31] Paolo Gatto and Jan S. Hesthaven, Numerical approximation of the fractional Laplacian via $ hp$-finite elements, with an application to image denoising, J. Sci. Comput. 65 (2015), no. 1, 249-270. MR 3394445,
  • [32] 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),
  • [33] V. Goldshtein and A. Ukhlov, Weighted Sobolev spaces and embedding theorems, Trans. Amer. Math. Soc. 361 (2009), no. 7, 3829-3850. MR 2491902 (2010b:46068),
  • [34] Jayadeep Gopalakrishnan and Joseph E. Pasciak, The convergence of V-cycle multigrid algorithms for axisymmetric Laplace and Maxwell equations, Math. Comp. 75 (2006), no. 256, 1697-1719 (electronic). MR 2240631 (2007g:65116),
  • [35] L. Greengard and V. Rokhlin, A fast algorithm for particle simulations, J. Comput. Phys. 73 (1987), no. 2, 325-348. MR 918448 (88k:82007),
  • [36] Michael Griebel, Karl Scherer, and Alexander Schweitzer, Robust norm equivalencies for diffusion problems, Math. Comp. 76 (2007), no. 259, 1141-1161 (electronic). MR 2299769 (2008d:65149),
  • [37] Wolfgang Hackbusch, Multigrid Methods and Applications, Springer Series in Computational Mathematics, vol. 4, Springer-Verlag, Berlin, 1985. MR 814495 (87e:65082)
  • [38] Wolfgang Hackbusch, The frequency decomposition multi-grid method. I. Application to anisotropic equations, Numer. Math. 56 (1989), no. 2-3, 229-245. MR 1018302 (90i:65212),
  • [39] W. Hackbusch, A sparse matrix arithmetic based on $ \mathcal {H}$-matrices. I. Introduction to $ \mathcal {H}$-matrices, Computing 62 (1999), no. 2, 89-108. MR 1694265 (2000c:65039),
  • [40] Helmut Harbrecht and Reinhold Schneider, Rapid solution of boundary integral equations by wavelet Galerkin schemes, Multiscale, nonlinear and adaptive approximation, Springer, Berlin, 2009, pp. 249-294. MR 2648376 (2011k:65170),
  • [41] Tuomas P. Hytönen, The sharp weighted bound for general Calderón-Zygmund operators, Ann. of Math. (2) 175 (2012), no. 3, 1473-1506. MR 2912709,
  • [42] N. S. Landkof, Foundations of Modern Potential Theory, translated from the Russian by A. P. Doohovskoy, Die Grundlehren der mathematischen Wissenschaften, Band 180, Springer-Verlag, New York-Heidelberg, 1972. MR 0350027 (50 #2520)
  • [43] Benjamin Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), 207-226. MR 0293384 (45 #2461)
  • [44] Ricardo H. Nochetto, Enrique Otárola, and Abner J. Salgado, A PDE approach to fractional diffusion in general domains: a priori error analysis, Found. Comput. Math. 15 (2015), no. 3, 733-791. MR 3348172,
  • [45] Ricardo H. Nochetto, Enrique Otárola, and Abner J. Salgado, Piecewise polynomial interpolation in Muckenhoupt weighted Sobolev spaces and applications, Numer. Math. 132 (2016), no. 1, 85-130. MR 3439216,
  • [46] 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),
  • [47] Elias M. Stein, Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals, with the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, III,, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. MR 1232192 (95c:42002)
  • [48] Rob Stevenson, Robustness of multi-grid applied to anisotropic equations on convex domains and on domains with re-entrant corners, Numer. Math. 66 (1993), no. 3, 373-398. MR 1246963 (94i:65047),
  • [49] Rob Stevenson, Adaptive wavelet methods for solving operator equations: an overview, Multiscale, nonlinear and adaptive approximation, Springer, Berlin, 2009, pp. 543-597. MR 2648381 (2011k:65196),
  • [50] Pablo Raúl Stinga and José Luis Torrea, Extension problem and Harnack's inequality for some fractional operators, Comm. Partial Differential Equations 35 (2010), no. 11, 2092-2122. MR 2754080 (2012c:35456),
  • [51] Bengt Ove Turesson, Nonlinear Potential Theory and Weighted Sobolev Spaces, Lecture Notes in Mathematics, vol. 1736, Springer-Verlag, Berlin, 2000. MR 1774162 (2002f:31027)
  • [52] Yongke Wu, Long Chen, Xiaoping Xie, and Jinchao Xu, Convergence analysis of V-cycle multigrid methods for anisotropic elliptic equations, IMA J. Numer. Anal. 32 (2012), no. 4, 1329-1347. MR 2991830,
  • [53] Jinchao Xu, Iterative methods by space decomposition and subspace correction, SIAM Rev. 34 (1992), no. 4, 581-613. MR 1193013 (93k:65029),
  • [54] Jinchao Xu, Long Chen, and Ricardo H. Nochetto, Optimal multilevel methods for $ H({\rm grad})$, $ H({\rm curl})$, and $ H({\rm div})$ systems on graded and unstructured grids, Multiscale, nonlinear and adaptive approximation, Springer, Berlin, 2009, pp. 599-659. MR 2648382 (2011k:65178),
  • [55] Jinchao Xu and Ludmil Zikatanov, The method of alternating projections and the method of subspace corrections in Hilbert space, J. Amer. Math. Soc. 15 (2002), no. 3, 573-597. MR 1896233 (2003f:65095),
  • [56] Chen-Song Zhang, Adaptive Finite Element Methods for Variational Inequalities: Theory and Applications In Finance, ProQuest LLC, Ann Arbor, MI. Thesis (Ph.D.)-University of Maryland, College Park, 2007. MR 2711028

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2010): 65N55, 65F10, 65N22, 65N30, 35S15, 65N12

Retrieve articles in all journals with MSC (2010): 65N55, 65F10, 65N22, 65N30, 35S15, 65N12

Additional Information

Long Chen
Affiliation: Department of Mathematics, University of California at Irvine, Irvine, California 92697

Ricardo H. Nochetto
Affiliation: Department of Mathematics and Institute for Physical Science and Technology, University of Maryland, College Park, Maryland 20742

Enrique Otárola
Affiliation: Departamento de Matemática, Universidad Técnica Federico Santa María, Valparaíso, Chile

Abner J. Salgado
Affiliation: Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996

Keywords: Finite elements, weighted Sobolev spaces, Muckenhoupt weights, anisotropic estimates, multilevel methods
Received by editor(s): March 17, 2014
Received by editor(s) in revised form: April 25, 2015
Published electronically: March 3, 2016
Additional Notes: The first author has been supported by NSF grants DMS-1115961, DMS-1418934, and DOE prime award # DE-SC0006903.
The second and fourth authors have been supported in part by NSF grants DMS-1109325 and DMS-1411808.
The third author was supported in part by the NSF grants DMS-1109325 and DMS-1411808 and by CONICYT through a CONICYT-FULBRIGHT Fellowship
The fourth author was supported by NSF grant DMS-1418784
Article copyright: © Copyright 2016 American Mathematical Society

American Mathematical Society