Quasi-optimal convergence rate for an adaptive method for the integral fractional Laplacian
HTML articles powered by AMS MathViewer
- by Markus Faustmann, Jens Markus Melenk and Dirk Praetorius;
- Math. Comp. 90 (2021), 1557-1587
- DOI: https://doi.org/10.1090/mcom/3603
- Published electronically: April 2, 2021
- HTML | PDF | Request permission
Abstract:
For the discretization of the integral fractional Laplacian $(-\Delta )^s$, $0 < s < 1$, based on piecewise linear functions, we present and analyze a reliable weighted residual a posteriori error estimator. In order to compensate for the lack of $L^2$-regularity of the residual in the regime $3/4 < s < 1$, this weighted residual error estimator includes as an additional weight a power of the distance from the mesh skeleton. We prove optimal convergence rates for an $h$-adaptive algorithm driven by this error estimator. Key to the analysis of the adaptive algorithm are local inverse estimates for the fractional Laplacian.References
- Gabriel Acosta, Francisco M. Bersetche, and Juan Pablo Borthagaray, A short FE implementation for a 2d homogeneous Dirichlet problem of a fractional Laplacian, Comput. Math. Appl. 74 (2017), no. 4, 784–816. MR 3679865, DOI 10.1016/j.camwa.2017.05.026
- 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
- Gabriel Acosta, Juan Pablo Borthagaray, and Norbert Heuer, Finite element approximations of the nonhomogeneous fractional Dirichlet problem, IMA J. Numer. Anal. 39 (2019), no. 3, 1471–1501. MR 4023752, DOI 10.1093/imanum/dry023
- Mark Ainsworth and Christian Glusa, Aspects of an adaptive finite element method for the fractional Laplacian: a priori and a posteriori error estimates, efficient implementation and multigrid solver, Comput. Methods Appl. Mech. Engrg. 327 (2017), 4–35. MR 3725761, DOI 10.1016/j.cma.2017.08.019
- Mark Ainsworth and Christian Glusa, Hybrid finite element–spectral method for the fractional Laplacian: approximation theory and efficient solver, SIAM J. Sci. Comput. 40 (2018), no. 4, A2383–A2405. MR 3835597, DOI 10.1137/17M1144696
- M. Aurada, M. Feischl, T. Führer, M. Karkulik, J. M. Melenk, and D. Praetorius, Local inverse estimates for non-local boundary integral operators, Math. Comp. 86 (2017), no. 308, 2651–2686. MR 3667020, DOI 10.1090/mcom/3175
- Markus Aurada, Michael Feischl, Thomas Führer, Michael Karkulik, and Dirk Praetorius, Energy norm based error estimators for adaptive BEM for hypersingular integral equations, Appl. Numer. Math. 95 (2015), 15–35. MR 3349683, DOI 10.1016/j.apnum.2013.12.004
- Lehel Banjai, Jens M. Melenk, Ricardo H. Nochetto, Enrique Otárola, Abner J. Salgado, and Christoph Schwab, Tensor FEM for spectral fractional diffusion, Found. Comput. Math. 19 (2019), no. 4, 901–962. MR 3989717, DOI 10.1007/s10208-018-9402-3
- Peter Binev, Wolfgang Dahmen, and Ron DeVore, Adaptive finite element methods with convergence rates, Numer. Math. 97 (2004), no. 2, 219–268. MR 2050077, DOI 10.1007/s00211-003-0492-7
- Peter Binev, Wolfgang Dahmen, Ronald DeVore, and Pencho Petrushev, Approximation classes for adaptive methods, Serdica Math. J. 28 (2002), no. 4, 391–416. Dedicated to the memory of Vassil Popov on the occasion of his 60th birthday. MR 1965238
- Andrea Bonito, Juan Pablo Borthagaray, Ricardo H. Nochetto, Enrique Otárola, and Abner J. Salgado, Numerical methods for fractional diffusion, Comput. Vis. Sci. 19 (2018), no. 5-6, 19–46. MR 3893441, DOI 10.1007/s00791-018-0289-y
- Andrea Bonito, Wenyu Lei, and Joseph E. Pasciak, Numerical approximation of the integral fractional Laplacian, Numer. Math. 142 (2019), no. 2, 235–278. MR 3941931, DOI 10.1007/s00211-019-01025-x
- Andrea Bonito and Joseph E. Pasciak, Numerical approximation of fractional powers of elliptic operators, Math. Comp. 84 (2015), no. 295, 2083–2110. MR 3356020, DOI 10.1090/S0025-5718-2015-02937-8
- Andrea Bonito and Joseph E. Pasciak, Numerical approximation of fractional powers of regularly accretive operators, IMA J. Numer. Anal. 37 (2017), no. 3, 1245–1273. MR 3671494, DOI 10.1093/imanum/drw042
- Juan Pablo Borthagaray and Patrick Ciarlet Jr., On the convergence in $H^1$-norm for the fractional Laplacian, SIAM J. Numer. Anal. 57 (2019), no. 4, 1723–1743. MR 3984304, DOI 10.1137/18M1221436
- Juan Pablo Borthagaray, Wenbo Li, and Ricardo H. Nochetto, Linear and nonlinear fractional elliptic problems, 75 years of mathematics of computation, Contemp. Math., vol. 754, Amer. Math. Soc., [Providence], RI, [2020] ©2020, pp. 69–92. MR 4132117, DOI 10.1090/conm/754/15145
- Xavier Cabré and Yannick Sire, Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates, Ann. Inst. H. Poincaré C Anal. Non Linéaire 31 (2014), no. 1, 23–53. MR 3165278, DOI 10.1016/j.anihpc.2013.02.001
- 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
- C. Carstensen, M. Feischl, M. Page, and D. Praetorius, Axioms of adaptivity, Comput. Math. Appl. 67 (2014), no. 6, 1195–1253. MR 3170325, DOI 10.1016/j.camwa.2013.12.003
- J. Manuel Cascon, Christian Kreuzer, Ricardo H. Nochetto, and Kunibert G. Siebert, Quasi-optimal convergence rate for an adaptive finite element method, SIAM J. Numer. Anal. 46 (2008), no. 5, 2524–2550. MR 2421046, DOI 10.1137/07069047X
- Long Chen, Ricardo H. Nochetto, Enrique Otárola, and Abner J. Salgado, A PDE approach to fractional diffusion: a posteriori error analysis, J. Comput. Phys. 293 (2015), 339–358. MR 3342475, DOI 10.1016/j.jcp.2015.01.001
- Willy Dörfler, A convergent adaptive algorithm for Poisson’s equation, SIAM J. Numer. Anal. 33 (1996), no. 3, 1106–1124. MR 1393904, DOI 10.1137/0733054
- Todd Dupont and Ridgway Scott, Polynomial approximation of functions in Sobolev spaces, Math. Comp. 34 (1980), no. 150, 441–463. MR 559195, DOI 10.1090/S0025-5718-1980-0559195-7
- Lawrence C. Evans, Partial differential equations, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 1998. MR 1625845, DOI 10.1090/gsm/019
- Birgit Faermann, Localization of the Aronszajn-Slobodeckij norm and application to adaptive boundary element methods. I. The two-dimensional case, IMA J. Numer. Anal. 20 (2000), no. 2, 203–234. MR 1752263, DOI 10.1093/imanum/20.2.203
- Birgit Faermann, Localization of the Aronszajn-Slobodeckij norm and application to adaptive boundary element methods. II. The three-dimensional case, Numer. Math. 92 (2002), no. 3, 467–499. MR 1930387, DOI 10.1007/s002110100319
- M. Faustmann, J.M. Melenk, and M. Parvizi, On the stability of the Scott-Zhang type operators and application to multilevel preconditioning in fractional diffusion, arXiv e-prints (2019-12).
- Michael Feischl, Thomas Führer, Michael Karkulik, Jens Markus Melenk, and Dirk Praetorius, Quasi-optimal convergence rates for adaptive boundary element methods with data approximation, part I: weakly-singular integral equation, Calcolo 51 (2014), no. 4, 531–562. MR 3279576, DOI 10.1007/s10092-013-0100-x
- Michael Feischl, Thomas Führer, Michael Karkulik, J. Markus Melenk, and Dirk Praetorius, Quasi-optimal convergence rates for adaptive boundary element methods with data approximation. Part II: Hyper-singular integral equation, Electron. Trans. Numer. Anal. 44 (2015), 153–176. MR 3313399
- M. Feischl, T. Führer, and D. Praetorius, Adaptive FEM with optimal convergence rates for a certain class of nonsymmetric and possibly nonlinear problems, SIAM J. Numer. Anal. 52 (2014), no. 2, 601–625. MR 3176325, DOI 10.1137/120897225
- M. Feischl, M. Karkulik, J. M. Melenk, and D. Praetorius, Quasi-optimal convergence rate for an adaptive boundary element method, SIAM J. Numer. Anal. 51 (2013), no. 2, 1327–1348. MR 3047442, DOI 10.1137/110842569
- Tsogtgerel Gantumur, Adaptive boundary element methods with convergence rates, Numer. Math. 124 (2013), no. 3, 471–516. MR 3066037, DOI 10.1007/s00211-013-0524-x
- Tsogtgerel Gantumur, Convergence rates of adaptive methods, Besov spaces, and multilevel approximation, Found. Comput. Math. 17 (2017), no. 4, 917–956. MR 3682217, DOI 10.1007/s10208-016-9308-x
- Fernando D. Gaspoz and Pedro Morin, Approximation classes for adaptive higher order finite element approximation, Math. Comp. 83 (2014), no. 289, 2127–2160. MR 3223327, DOI 10.1090/S0025-5718-2013-02777-9
- P. Grisvard, Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics, vol. 24, Pitman (Advanced Publishing Program), Boston, MA, 1985. MR 775683
- Gerd Grubb, Fractional Laplacians on domains, a development of Hörmander’s theory of $\mu$-transmission pseudodifferential operators, Adv. Math. 268 (2015), 478–528. MR 3276603, DOI 10.1016/j.aim.2014.09.018
- Michael Karkulik and Jens Markus Melenk, $\mathcal H$-matrix approximability of inverses of discretizations of the fractional Laplacian, Adv. Comput. Math. 45 (2019), no. 5-6, 2893–2919. MR 4047021, DOI 10.1007/s10444-019-09718-5
- Michael Karkulik, David Pavlicek, and Dirk Praetorius, On 2D newest vertex bisection: optimality of mesh-closure and $H^1$-stability of $L_2$-projection, Constr. Approx. 38 (2013), no. 2, 213–234. MR 3097045, DOI 10.1007/s00365-013-9192-4
- Mateusz Kwaśnicki, Ten equivalent definitions of the fractional Laplace operator, Fract. Calc. Appl. Anal. 20 (2017), no. 1, 7–51. MR 3613319, DOI 10.1515/fca-2017-0002
- A. Lischke, G. Pang, M. Gulian, F. Song, C. Glusa, X. Zheng, Z. Mao, W. Cai, M.M. Meerschaert, M. Ainsworth, and G.E. Karniadakis, What is the fractional Laplacian? A comparative review with new results, J. Comput. Phys. 404 (2020), 109009, 62.
- William McLean, Strongly elliptic systems and boundary integral equations, Cambridge University Press, Cambridge, 2000. MR 1742312
- 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, DOI 10.1007/s10208-014-9208-x
- Ricardo H. Nochetto, Tobias von Petersdorff, and Chen-Song Zhang, A posteriori error analysis for a class of integral equations and variational inequalities, Numer. Math. 116 (2010), no. 3, 519–552. MR 2684296, DOI 10.1007/s00211-010-0310-y
- L. Ridgway Scott and Shangyou Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions, Math. Comp. 54 (1990), no. 190, 483–493. MR 1011446, DOI 10.1090/S0025-5718-1990-1011446-7
- Fangying Song, Chuanju Xu, and George Em Karniadakis, Computing fractional Laplacians on complex-geometry domains: algorithms and simulations, SIAM J. Sci. Comput. 39 (2017), no. 4, A1320–A1344. MR 3679919, DOI 10.1137/16M1078197
- Rob Stevenson, Optimality of a standard adaptive finite element method, Found. Comput. Math. 7 (2007), no. 2, 245–269. MR 2324418, DOI 10.1007/s10208-005-0183-0
- Rob Stevenson, The completion of locally refined simplicial partitions created by bisection, Math. Comp. 77 (2008), no. 261, 227–241. MR 2353951, DOI 10.1090/S0025-5718-07-01959-X
- WenYi Tian, Han Zhou, and Weihua Deng, A class of second order difference approximations for solving space fractional diffusion equations, Math. Comp. 84 (2015), no. 294, 1703–1727. MR 3335888, DOI 10.1090/S0025-5718-2015-02917-2
- Xuan Zhao, Xiaozhe Hu, Wei Cai, and George Em Karniadakis, Adaptive finite element method for fractional differential equations using hierarchical matrices, Comput. Methods Appl. Mech. Engrg. 325 (2017), 56–76. MR 3693419, DOI 10.1016/j.cma.2017.06.017
Bibliographic Information
- Markus Faustmann
- Affiliation: TU Wien, Institute of Analysis und Scientific Computing, Wiedner Hauptstraße 8-10, 1040 Wien, Austria
- MR Author ID: 1123286
- Email: markus.faustmann@tuwien.ac.at
- Jens Markus Melenk
- Affiliation: TU Wien, Institute of Analysis und Scientific Computing, Wiedner Hauptstraße 8-10, 1040 Wien, Austria
- MR Author ID: 613978
- ORCID: 0000-0001-9024-6028
- Email: melenk@tuwien.ac.at
- Dirk Praetorius
- Affiliation: TU Wien, Institute of Analysis und Scientific Computing, Wiedner Hauptstraße 8-10, 1040 Wien, Austria
- MR Author ID: 702616
- ORCID: 0000-0002-1977-9830
- Email: dirk.praetorius@tuwien.ac.at
- Received by editor(s): March 25, 2019
- Received by editor(s) in revised form: April 15, 2020, and August 28, 2020
- Published electronically: April 2, 2021
- Additional Notes: The research of the second and third authors was funded by the Austrian Science Fund (FWF) by the special research program Taming complexity in PDE systems (grant SFB F65). Additionally, the third author acknowledges support through the FWF research project Optimal adaptivity for BEM and FEM-BEM coupling (grant P27005).
- © Copyright 2021 American Mathematical Society
- Journal: Math. Comp. 90 (2021), 1557-1587
- MSC (2020): Primary 65N30, 65N50, 35R11
- DOI: https://doi.org/10.1090/mcom/3603
- MathSciNet review: 4273109