Artículo

Estamos trabajando para incorporar este artículo al repositorio
Consulte el artículo en la página del editor
Consulte la política de Acceso Abierto del editor

Abstract:

The purpose of this work is to study a finite element method for finding solutions to the eigenvalue problem for the fractional Laplacian. We prove that the discrete eigenvalue problem converges to the continuous one and we show the order of such convergence. Finally, we perform some numerical experiments and compare our results with previous work by other authors. © 2018, Springer Science+Business Media, LLC, part of Springer Nature.

Registro:

Documento: Artículo
Título:Finite Element Approximation for the Fractional Eigenvalue Problem
Autor:Borthagaray, J.P.; Del Pezzo, L.M.; Martínez, S.
Filiación:IMAS - CONICET and Departamento de Matemática, FCEyN - Universidad de Buenos Aires, Ciudad Universitaria, Pabellón I (1428), Buenos Aires, Argentina
CONICET and UTDT, Departamento de Matemáticas y Estadística, Universidad Torcuato Di Tella, Av. Figueroa Alcorta 7350, Buenos Aires, C1428BCW, Argentina
Palabras clave:Eigenvalue problem; Finite element method; Fractional Laplacian; Finite element method; Laplace transforms; Eigenvalue problem; Finding solutions; Finite element approximations; Fractional Laplacian; Numerical experiments; Eigenvalues and eigenfunctions
Año:2018
Volumen:77
Número:1
Página de inicio:308
Página de fin:329
DOI: http://dx.doi.org/10.1007/s10915-018-0710-1
Título revista:Journal of Scientific Computing
Título revista abreviado:J Sci Comput
ISSN:08857474
CODEN:JSCOE
Registro:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_08857474_v77_n1_p308_Borthagaray

Referencias:

  • Acosta, G., Bersetche, F., Borthagaray, J.P., A short FEM implementation for a 2d homogeneous Dirichlet problem of a fractional Laplacian (2017) Comput. Math. Appl., 74 (4), pp. 784-816
  • Acosta, G., Borthagaray, J.P., A fractional Laplace equation: regularity of solutions and finite element approximations (2017) SIAM J. Numer. Anal., 55 (2), pp. 472-495
  • Ainsworth, M., Glusa, C., (2017) Towards an Efficient Finite Element Method for the Integral Fractional Laplacian on Polygonal Domains
  • Amore, P., Fernández, F.M., Hofmann, C.P., Sáenz, R.A., Collocation method for fractional quantum mechanics (2010) J. Math. Phys., 51 (12), p. 122101
  • Antoine, X., Tang, Q., Zhang, Y., On the ground states and dynamics of space fractional nonlinear Schrödinger/Gross-Pitaevskii equations with rotation term and nonlocal nonlinear interactions (2016) J. Comput. Phys., 325, pp. 74-97
  • Babuška, I., Osborn, J., Eigenvalue problems (1991) Handbook of Numerical Analysis, 2, pp. 641-787. , Handb. Numer. Anal., II, North-Holland, Amsterdam
  • Bao, W., Ruan, X., Shen, J., Sheng, C., (2018) Fundamental gaps of the fractional Schrödinger operator
  • Benson, D.A., Wheatcraft, S.W., Meerschaert, M.M., Application of a fractional advection–dispersion equation (2000) Water Resour. Res., 36 (6), pp. 1403-1412
  • Bertoin, J., (1996) Lévy Processes, Volume 121 of Cambridge Tracts in Mathematics, , Cambridge University Press, Cambridge
  • Boffi, D., Finite element approximation of eigenvalue problems (2010) Acta Numer., 19, pp. 1-120
  • Boffi, D., Brezzi, F., Gastaldi, L., On the problem of spurious eigenvalues in the approximation of linear elliptic problems in mixed form (2000) Math. Comput., 69 (229), pp. 121-140
  • Bonito, A., Borthagaray, J.P., Nochetto, R.H., Otárola, E., Salgado, A.J., Numerical methods for fractional diffusion (2018) Computing and Visualization in Science
  • Buades, A., Coll, B., Morel, J.M., Image denoising methods. A new nonlocal principle (2010) SIAM Rev., 52 (1), pp. 113-147. , Reprint of A review of image denoising algorithms, with a new one [MR2162865]
  • Caffarelli, L., Silvestre, L., Regularity theory for fully nonlinear integro-differential equations (2009) Commun. Pure Appl. Math., 62 (5), pp. 597-638
  • Carr, P., Geman, H., Madan, D.B., Yor, M., The fine structure of asset returns: an empirical investigation (2002) J. Bus., 75 (2), pp. 305-332
  • Chen, Z.-Q., Song, R., Two-sided eigenvalue estimates for subordinate processes in domains (2005) J. Funct. Anal., 226 (1), pp. 90-113
  • Clément, P., Approximation by finite element functions using local regularization (1975) Rev. Française Automat. Informat. Recherche Opérationnelle Sér. RAIRO Analyse Numérique, 9 (R–2), pp. 77-84
  • Cont, R., Tankov, P., (2004) Financial Modelling with Jump Processes, , Chapman & Hall/CRC Financial Mathematics Series, Chapman & Hall, Boca Raton
  • Cushman, J.H., Ginn, T.R., Nonlocal dispersion in media with continuously evolving scales of heterogeneity (1993) Transp. Porous Media, 13 (1), pp. 123-138
  • Del Pezzo, L.M., Quaas, A., Global bifurcation for fractional p -Laplacian and an application (2016) Z. Anal. Anwend., 35 (4), pp. 411-447
  • Demengel, F., Demengel, G., (2012) Functional Spaces for the Theory of Elliptic Partial Differential Equations, , Universitext. Springer, London; EDP Sciences, Les Ulis, Translated from the 2007 French original by Reinie Erné
  • Di Nezza, E., Palatucci, G., Valdinoci, E., Hitchhiker’s guide to the fractional Sobolev spaces (2012) Bull. Sci. Math., 136 (5), pp. 521-573
  • Duo, S., Zhang, Y., Computing the ground and first excited states of the fractional Schrödinger equation in an infinite potential well (2015) Commun. Comput. Phys., 18 (2), pp. 321-350
  • Dyda, B., Kuznetsov, A., Kwaśnicki, M., Eigenvalues of the fractional Laplace operator in the unit ball (2017) J. Lond. Math. Soc., 95 (2), pp. 500-518
  • Dyda, B., Kuznetsov, A., Kwaśnicki, M., Fractional Laplace operator and Meijer G-function (2017) Constr. Approx., 45 (3), pp. 427-448
  • Gatto, P., Hesthaven, J.S., Numerical approximation of the fractional Laplacian via hp-finite elements, with an application to image denoising (2015) J. Sci. Comp., 65 (1), pp. 249-270
  • Ghelardoni, P., Magherini, C., A matrix method for fractional Sturm–Liouville problems on bounded domain (2017) Adv. Comput. Math., 43 (6), pp. 1377-1401
  • Gilboa, G., Osher, S., Nonlocal operators with applications to image processing (2008) Multiscale Model. Simul., 7 (3), pp. 1005-1028
  • Grisvard, P., (1985) Elliptic Problems in Nonsmooth Domains, 24. , Pitman (Advanced Publishing Program), Boston, MA
  • Grubb, G., Fractional laplacians on domains, a development of Hörmander’s theory of μ -transmission pseudodifferential operators (2015) Adv. Math., 268, pp. 478-528
  • Grubb, G., Spectral results for mixed problems and fractional elliptic operators (2015) J. Math. Anal. Appl., 421 (2), pp. 1616-1634
  • Kato, T., (1995) Perturbation Theory for Linear Operators, , Classics in Mathematics. Springer, Berlin, Reprint of the 1980 edition
  • Klafter, J., Sokolov, I.M., Anomalous diffusion spreads its wings (2005) Phys. World, 18 (8), p. 29
  • Kufner, A., (1985) Weighted Sobolev Spaces. a Wiley-Interscience Publication, , Wiley, New York, Translated from the Czech
  • Kulczycki, T., Kwaśnicki, M., Małecki, J., Stos, A., Spectral properties of the Cauchy process on half-line and interval (2010) Proc. Lond. Math. Soc. (3), 101 (2), pp. 589-622
  • Kwaśnicki, M., Eigenvalues of the fractional Laplace operator in the interval (2012) J. Funct. Anal., 262 (5), pp. 2379-2402
  • Laskin, N., Fractional Schrödinger equation (2002) Phys. Rev. E, 66 (5), p. 056108
  • Lions, J.L., Magenes, E., (2012) Non-Homogeneous Boundary Value Problems and Applications, 1. , Springer, Berlin
  • Lou, Y., Zhang, X., Osher, S., Bertozzi, A., Image recovery via nonlocal operators (2010) J. Sci. Comput., 42 (2), pp. 185-197
  • Luchko, Y., Fractional Schrödinger equation for a particle moving in a potential well (2013) J. Math. Phys., 54 (1), p. 012111
  • McCay, B.M., Narasimhan, M.N.L., Theory of nonlocal electromagnetic fluids (1981) Arch. Mech. (Arch. Mech. Stos.), 33 (3), pp. 365-384
  • Metzler, R., Klafter, J., The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics (2004) J. Phys. A, 37 (31), pp. R161-R208
  • Raviart, P.-A., Thomas, J.-M., Introduction à l’analyse numérique des équations aux dérivées partielles (1983) Collection Mathématiques Appliquées Pour La Maîtrise, , Collection of Applied Mathematics for the Master’s Degree, Masson, Paris
  • Ros-Oton, X., Serra, J., The Dirichlet problem for the fractional Laplacian: regularity up to the boundary (2014) Journal de Mathématiques Pures et Appliquées, 101 (3), pp. 275-302
  • Ros-Oton, X., Serra, J., Local integration by parts and Pohozaev identities for higher order fractional Laplacians (2015) Discrete Contin. Dyn. Syst., 35 (5), pp. 2131-2150
  • Scott, L.R., Zhang, S., Finite element interpolation of nonsmooth functions satisfying boundary conditions (1990) Math. Comput., 54 (190), pp. 483-493
  • Servadei, R., The Yamabe equation in a non-local setting (2013) Adv. Nonlinear Anal., 2 (3), pp. 235-270
  • Servadei, R., Valdinoci, E., A Brezis–Nirenberg result for non-local critical equations in low dimension (2013) Commun. Pure Appl. Anal., 12 (6), pp. 2445-2464
  • Servadei, R., Valdinoci, E., Variational methods for non-local operators of elliptic type (2013) Discrete Contin. Dyn. Syst., 33 (5), pp. 2105-2137
  • Servadei, R., Valdinoci, E., On the spectrum of two different fractional operators (2014) Proc. R. Soc. Edinburgh Sect. A, 144 (4), pp. 831-855
  • Servadei, R., Valdinoci, E., Weak and viscosity solutions of the fractional Laplace equation (2014) Publ. Mat., 58 (1), pp. 133-154
  • Servadei, R., Valdinoci, E., The Brezis–Nirenberg result for the fractional Laplacian (2015) Trans. Am. Math. Soc., 367 (1), pp. 67-102
  • Silling, S.A., Reformulation of elasticity theory for discontinuities and long-range forces (2000) J. Mech. Phys. Solids, 48 (1), pp. 175-209
  • Silvestre, L., Regularity of the obstacle problem for a fractional power of the Laplace operator (2007) Commun. Pure Appl. Math., 60 (1), pp. 67-112
  • Valdinoci, E., From the long jump random walk to the fractional Laplacian (2009) Bol. Soc. Esp. Mat. Apl. S E → MA, 49, pp. 33-44
  • Zhou, Y., Fractional Sobolev extension and imbedding (2015) Trans. Am. Math. Soc., 367 (2), pp. 959-979
  • Zoia, A., Rosso, A., Kardar, M., Fractional Laplacian in bounded domains (2007) Phys. Rev. E (3), 76 (2), p. 021116, 11

Citas:

---------- APA ----------
Borthagaray, J.P., Del Pezzo, L.M. & Martínez, S. (2018) . Finite Element Approximation for the Fractional Eigenvalue Problem. Journal of Scientific Computing, 77(1), 308-329.
http://dx.doi.org/10.1007/s10915-018-0710-1
---------- CHICAGO ----------
Borthagaray, J.P., Del Pezzo, L.M., Martínez, S. "Finite Element Approximation for the Fractional Eigenvalue Problem" . Journal of Scientific Computing 77, no. 1 (2018) : 308-329.
http://dx.doi.org/10.1007/s10915-018-0710-1
---------- MLA ----------
Borthagaray, J.P., Del Pezzo, L.M., Martínez, S. "Finite Element Approximation for the Fractional Eigenvalue Problem" . Journal of Scientific Computing, vol. 77, no. 1, 2018, pp. 308-329.
http://dx.doi.org/10.1007/s10915-018-0710-1
---------- VANCOUVER ----------
Borthagaray, J.P., Del Pezzo, L.M., Martínez, S. Finite Element Approximation for the Fractional Eigenvalue Problem. J Sci Comput. 2018;77(1):308-329.
http://dx.doi.org/10.1007/s10915-018-0710-1