Una Formulación Mixta para el Problema de Poisson Fraccionario
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https://doi.org/10.70567/mc.v41i15.77Palabras clave:
Laplaciano fraccionario, formulación mixta, método de elementos finitosResumen
La formulación mixta del problema de Poisson clásico consiste en introducir un flujo como nueva variable con condiciones de borde adecuadas, obteniendo un sistema de ecuaciones acopladas. Usando identidades del cálculo fraccionario, en este trabajo exploramos una formulación mixta del problema de Poisson fraccionario y probamos que el problema está bien planteado. Una discretización directa del problema no es posible, por lo que siguiendo ideas de Hughes y Masud introducimos una formulación estabilizada, que da lugar a un problema coercivo y bien planteado. La coercividad implica que cualquier discretización por elementos finitos conforme sea estable. Por último, obtenemos la convergencia de estas discretizaciones y discutimos su implementación.
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Acosta G. and Borthagaray J.P. A fractional laplace equation: regularity of solutions and finite element approximations. SIAM Journal on Numerical Analysis, 55(2):472-495, 2017. https://doi.org/10.1137/15M1033952
Boffi D., Brezzi F., and Fortin M. Mixed finite element methods and applications, volume 44 of Springer Ser. Comput. Math. Berlin: Springer, 2013. ISBN 978-3-642-36518-8; 978-3-642-36519-5.
Borthagaray J.P. and Nochetto R.H. Besov regularity for the Dirichlet integral fractional Laplacian in Lipschitz domains. J. Funct. Anal., 284(6):33, 2023. ISSN 0022-1236. https://doi.org/10.1016/j.jfa.2022.109829
Borthagaray J.P., Nochetto R.H., and Salgado A.J. Weighted sobolev regularity and rate of approximation of the obstacle problem for the integral fractional laplacian. 2019. https://doi.org/10.1142/S021820251950057X
Buades A., Coll B., and Morel J.M. Image denoising methods. a new nonlocal principle. SIAM review, 52(1):113-147, 2010. https://doi.org/10.1137/090773908
Carr P., Geman H., Madan D., and Yor M. The fine structure of asset returns: An empirical investigation. The Journal of Business, 75(2):305-332, 2002. ISSN 00219398, 15375374. https://doi.org/10.1086/338705
Chen Z. and Nochetto R.H. Residual type a posteriori error estimates for elliptic obstacle problems. Numerische Mathematik, 84:527-548, 2000. https://doi.org/10.1007/s002110050009
Comi G.E. and Stefani G. A distributional approach to fractional Sobolev spaces and fractional variation: existence of blow-up. J. Funct. Anal., 277(10):3373-3435, 2019. ISSN 0022-1236. https://doi.org/10.1016/j.jfa.2019.03.011
Constantin P. andWu J. Behavior of solutions of 2d quasi-geostrophic equations. SIAM journal on mathematical analysis, 30(5):937-948, 1999. https://doi.org/10.1137/S0036141098337333
Daoud M. and Laamri E.H. Fractional laplacians: A short survey. Discrete & Continuous Dynamical Systems-S, 15(1):95-116, 2022. https://doi.org/10.3934/dcdss.2021027
De Pablo A., Quirós F., Rodríguez A., and Vázquez J.L. A general fractional porous medium equation. Communications on Pure and Applied Mathematics, 65(9):1242-1284, 2012. https://doi.org/10.1002/cpa.21408
D'Elia M., Gulian M., Mengesha T., and Scott J.M. Connections between nonlocal operators: from vector calculus identities to a fractional helmholtz decomposition. 2021a. https://doi.org/10.2172/1855046
D'Elia M., Gulian M., Olson H., and Karniadakis G.E. Towards a unified theory of fractional and nonlocal vector calculus. 2021b. https://doi.org/10.2172/1841821
Di Nezza E., Palatucci G., and Valdinoci E. Hitchhiker's guide to the fractional sobolev spaces. Bulletin des sciences mathématiques, 136(5):521-573, 2012. https://doi.org/10.1016/j.bulsci.2011.12.004
Edmunds D.E. and EvansW.D. Fractional Sobolev spaces and inequalities, volume 230. Cambridge University Press, 2022. https://doi.org/10.1017/9781009254625
Gilboa G. and Osher S. Nonlocal linear image regularization and supervised segmentation. Multiscale Modeling & Simulation, 6(2):595-630, 2007. https://doi.org/10.1137/060669358
Lischke A., Pang G., Gulian M., Song F., Glusa C., Zheng X., Mao Z., Cai W., Meerschaert M.M., Ainsworth M., and Karniadakis G.E. What is the fractional laplacian? 2019.
Lou Y., Zhang X., Osher S., and Bertozzi A. Image recovery via nonlocal operators. Journal of Scientific Computing, 42(2):185-197, 2010. https://doi.org/10.1007/s10915-009-9320-2
Lu F., An Q., and Yu Y. Nonparametric learning of kernels in nonlocal operators. 2022.
Masud A. and Hughes T.J.R. A stabilized mixed finite element method for Darcy flow. Comput. Methods Appl. Mech. Eng., 191(39-40):4341-4370, 2002. ISSN 0045-7825. https://doi.org/10.1016/S0045-7825(02)00371-7
Rosasco L., Belkin M., and Vito E.D. On learning with integral operators. Journal of Machine Learning Research, 11(30):905-934, 2010.
Schekochihin A.A., Cowley S.C., and Yousef T.A. Mhd turbulence: Nonlocal, anisotropic, nonuniversal? In IUTAM Symposium on Computational Physics and New Perspectives in Turbulence: Proceedings of the IUTAM Symposium on Computational Physics and New Perspectives in Turbulence, Nagoya University, Nagoya, Japan, September, 11-14, 2006, pages 347-354. Springer, 2008. https://doi.org/10.1007/978-1-4020-6472-2_52
Shieh T.T. and Spector D.E. On a new class of fractional partial differential equations. Adv. Calc. Var., 8(4):321-336, 2015. ISSN 1864-8258. https://doi.org/10.1515/acv-2014-0009
Treeby B.E. and Cox B.T. Modeling power law absorption and dispersion for acoustic propagation using the fractional laplacian. The Journal of the Acoustical Society of America, 127(5):2741-2748, 2010. https://doi.org/10.1121/1.3377056
Wei Y., Kang Y., Yin W., and Wang Y. Generalization of the gradient method with fractional order gradient direction. Journal of the Franklin Institute, 357(4):2514-2532, 2020. https://doi.org/10.1016/j.jfranklin.2020.01.008
Yamamoto M. Asymptotic expansion of solutions to the dissipative equation with fractional laplacian. SIAM Journal on Mathematical Analysis, 44(6):3786-3805, 2012. https://doi.org/10.1137/120873200
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Derechos de autor 2024 Asociación Argentina de Mecánica Computacional
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