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Discontinuous Galerkin methods for fractional elliptic problems. (English) Zbl 07195819
Summary: The aim of this paper is to provide a mathematical framework for studying different versions of discontinuous Galerkin (DG) approaches for solving 2D Riemann-Liouville fractional elliptic problems on a finite domain. The boundedness and stability analysis of the primal bilinear form are provided. A priori error estimate under energy norm and optimal error estimate under \(L^2\) norm are obtained for DG methods of the different formulations. Finally, the performed numerical examples confirm the optimal convergence order of the different formulations.
MSC:
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35R11 Fractional partial differential equations
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[1] Aboelenen, T., Local discontinuous Galerkin method for distributed-order time and space-fractional convection-diffusion and Schrödinger type equations, Nonlinear Dyn, 92, 2, 395-413 (2018) · Zbl 1398.35261
[2] Aboelenen, T., A direct discontinuous Galerkin method for fractional convection-diffusion and Schrödinger-type equations, Eur Phys J Plus, 133, 8, 316 (2018)
[3] Aboelenen, T., A high-order nodal discontinuous Galerkin method for nonlinear fractional Schrödinger type equations, Commun Nonlinear Sci Numer Simul, 54, 428-452 (2018)
[4] Aboelenen, T.; El-Hawary, H., A high-order nodal discontinuous Galerkin method for a linearized fractional Cahn-Hilliard equation, Comput Math Appl, 73, 6, 1197-1217 (2017) · Zbl 1412.65132
[5] Aboelenen, T.; Bakr, S.; El-Hawary, H., Fractional Laguerre spectral methods and their applications to fractional differential equations on unbounded domain, Int J Comput Math (2015) · Zbl 1375.65135
[6] Adams, Ra, Sobolev spaces (1975), New York: Academic Press, New York
[7] Arnold, Dn, An interior penalty finite element method with discontinuous elements, SIAM J Numer Anal, 19, 4, 742-760 (1982) · Zbl 0482.65060
[8] Arnold, Dn; Brezzi, F.; Cockburn, B.; Marini, D., Discontinuous Galerkin methods for elliptic problems, Lect Notes Comput Sci Eng, 11, 89-102 (2000)
[9] Arnold, Dn; Brezzi, F.; Cockburn, B.; Marini, Ld, Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J Numer Anal, 39, 5, 1749-1779 (2002) · Zbl 1008.65080
[10] Baker, Ga, Finite element methods for elliptic equations using nonconforming elements, Math Comput, 31, 137, 45-59 (1977) · Zbl 0364.65085
[11] Bassi, F.; Rebay, S., A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations, J Comput Phys, 131, 2, 267-279 (1997) · Zbl 0871.76040
[12] Baumann, Ce; Oden, Jt, A discontinuous hp finite element method for convection-diffusion problems, Comput Methods Appl Mech Eng, 175, 3-4, 311-341 (1999) · Zbl 0924.76051
[13] Brenner, Sc; Owens, L.; Sung, Ly, A weakly over-penalized symmetric interior penalty method, Electron Trans Numer Anal, 30, 107-127 (2008) · Zbl 1171.65077
[14] Castillo, P.; Cockburn, B.; Karniadakis, Ge; Shu, Cw, An optimal estimate for the local discontinuous Galerkin method, Discontinuous Galerkin methods. Lecture notes in computational science and engineering, 285-290 (2000), Berlin, Heidelberg: Springer, Berlin, Heidelberg
[15] Castillo, P.; Cockburn, B.; Perugia, I.; Schötzau, D., An a priori error analysis of the local discontinuous Galerkin method for elliptic problems, SIAM J Numer Anal, 38, 5, 1676-1706 (2000) · Zbl 0987.65111
[16] Cockburn, Bernardo, Discontinuous Galerkin Methods for Convection-Dominated Problems, High-Order Methods for Computational Physics, 69-224 (1999), Berlin, Heidelberg: Springer Berlin Heidelberg, Berlin, Heidelberg · Zbl 0937.76049
[17] Cockburn, B.; Dawson, C., Approximation of the velocity by coupling discontinuous Galerkin and mixed finite element methods for flow problems, Comput Geosci, 6, 3-4, 505-522 (2002) · Zbl 1023.76020
[18] Cockburn, B.; Shu, Cw, TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II. General framework, Math Comput, 52, 186, 411-435 (1989) · Zbl 0662.65083
[19] Cockburn, B.; Kanschat, G.; Schötzau, D., A locally conservative LDG method for the incompressible Navier-Stokes equations, Math Comput, 74, 251, 1067-1095 (2005) · Zbl 1069.76029
[20] Cockburn, B.; Gopalakrishnan, J.; Lazarov, R., Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems, SIAM J Numer Anal, 47, 2, 1319-1365 (2009) · Zbl 1205.65312
[21] Cuesta, E.; Kirane, M.; Malik, Sa, Image structure preserving denoising using generalized fractional time integrals, Signal Process, 92, 2, 553-563 (2012)
[22] Deng, W., Finite element method for the space and time fractional Fokker-Planck equation, SIAM J Numer Anal, 47, 1, 204-226 (2009) · Zbl 1416.65344
[23] Deng, W.; Hesthaven, Js, Local discontinuous Galerkin methods for fractional diffusion equations, ESAIM Math Model Numer Anal, 47, 6, 1845-1864 (2013) · Zbl 1282.35400
[24] Ervin, Vj; Roop, Jp, Variational formulation for the stationary fractional advection dispersion equation, Numer Methods Partial Differ Equ, 22, 3, 558-576 (2006) · Zbl 1095.65118
[25] Ervin, Vj; Heuer, N.; Roop, Jp, Numerical approximation of a time dependent, nonlinear, space fractional diffusion equation, SIAM J Numer Anal, 45, 2, 572-591 (2007) · Zbl 1141.65089
[26] Golbabai, A.; Nikan, O.; Nikazad, T., Numerical analysis of time fractional Black-Scholes european option pricing model arising in financial market, Comput Appl Math, 38, 4, 173 (2019) · Zbl 07114279
[27] Golbabai, A.; Nikan, O.; Nikazad, T., Numerical investigation of the time fractional mobile-immobile advection-dispersion model arising from solute transport in porous media, Int J Appl Comput Math, 5, 3, 50 (2019) · Zbl 1411.76113
[28] Hesthaven, Js; Warburton, T., Nodal discontinuous Galerkin methods: algorithms, analysis, and applications (2007), Berlin: Springer Science & Business Media, Berlin
[29] Ji, X.; Tang, H., High-order accurate Runge-Kutta (local) discontinuous Galerkin methods for one-and two-dimensional fractional diffusion equations, Numer Math Theory Methods Appl, 5, 3, 333-358 (2012) · Zbl 1274.65271
[30] Jin, B.; Lazarov, R.; Pasciak, J.; Zhou, Z., Error analysis of a finite element method for the space-fractional parabolic equation, SIAM J Numer Anal, 52, 5, 2272-2294 (2014) · Zbl 1310.65126
[31] Kharazmi, E.; Zayernouri, M.; Karniadakis, Ge, A Petrov-Galerkin spectral element method for fractional elliptic problems, Comput Methods Appl Mech Eng, 324, 512-536 (2017)
[32] Kilbas, Anatoly A.; Srivastava, Hari M.; Trujillo, Juan J., Preface, Theory and Applications of Fractional Differential Equations, vii-x (2006)
[33] Kilbas, A.; Marichev, O.; Samko, S., Fractional integral and derivatives (theory and applications), Gordon Breach Switz, 1, 993, 1 (1993)
[34] Li, X.; Xu, C., A space-time spectral method for the time fractional diffusion equation, SIAM J Numer Anal, 47, 3, 2108-2131 (2009) · Zbl 1193.35243
[35] Magin, Rl, Fractional calculus in bioengineering (2006), Redding: Begell House, Redding
[36] Mainardi, F., Fractional calculus and waves in linear viscoelasticity: an introduction to mathematical models (2010), Singapore: World Scientific, Singapore · Zbl 1210.26004
[37] Meerschaert, Mm; Tadjeran, C., Finite difference approximations for fractional advection-dispersion flow equations, J Comput Appl Math, 172, 1, 65-77 (2004) · Zbl 1126.76346
[38] Meerschaert, Mm; Scheffler, Hp; Tadjeran, C., Finite difference methods for two-dimensional fractional dispersion equation, J Comput Phys, 211, 1, 249-261 (2006) · Zbl 1085.65080
[39] Metzler, R.; Klafter, J., The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics, J Phys A Math Gen, 37, 31, R161 (2004) · Zbl 1075.82018
[40] Miller K, Ross B (1993) An Introduction to the fractional calculus and fractional differential equations. Wiley, Hoboken. https://books.google.co.in/books?id=MOp_QgAACAAJ · Zbl 0789.26002
[41] Moffatt, Hk; Zaslavsky, G.; Comte, P.; Tabor, M., Topological aspects of the dynamics of fluids and plasmas (2013), Berlin: Springer Science & Business Media, Berlin
[42] Mustapha, K.; Mclean, W., Piecewise-linear, discontinuous Galerkin method for a fractional diffusion equation, Numer Algorithms, 56, 2, 159-184 (2011) · Zbl 1211.65127
[43] Mustapha, K.; Mclean, W., Uniform convergence for a discontinuous Galerkin, time-stepping method applied to a fractional diffusion equation, IMA J Numer Anal, 32, 3, 906-925 (2012) · Zbl 1327.65177
[44] Mustapha, K.; Mclean, W., Superconvergence of a discontinuous Galerkin method for fractional diffusion and wave equations, SIAM J Numer Anal, 51, 1, 491-515 (2013) · Zbl 1267.26005
[45] Nikan, O.; Machado, Jt; Golbabai, A.; Nikazad, T., Numerical investigation of the nonlinear modified anomalous diffusion process, Nonlinear Dyn, 97, 4, 2757-2775 (2019) · Zbl 1430.60091
[46] Nikan O, Golbabai A, Machado JT, Nikazad T (2020) Numerical solution of the fractional rayleigh-stokes model arising in a heated generalized second-grade fluid. Eng Comput. 10.1007/s00366-019-00913-y
[47] Oden, Jt; Babuŝka, I.; Baumann, Ce, A discontinuous hp finite element method for diffusion problems, J Comput Phys, 146, 2, 491-519 (1998) · Zbl 0926.65109
[48] Oldham, Kb, Fractional differential equations in electrochemistry, Adv Eng Softw, 41, 1, 9-12 (2010) · Zbl 1177.78041
[49] Peraire, J.; Persson, Po, The compact discontinuous Galerkin (CDG) method for elliptic problems, SIAM J Sci Comput, 30, 4, 1806-1824 (2008) · Zbl 1167.65436
[50] Podlubny, I., Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications (1998), Cambridge: Academic press, Cambridge · Zbl 0922.45001
[51] Qiu, L.; Deng, W.; Hesthaven, Js, Nodal discontinuous Galerkin methods for fractional diffusion equations on 2D domain with triangular meshes, J Comput Phys, 298, 678-694 (2015) · Zbl 1349.65476
[52] Qiu, L.; Deng, W.; Hesthaven, Js, Nodal discontinuous Galerkin methods for fractional diffusion equations on 2D domain with triangular meshes, J Comput Phys, 298, 678-694 (2015) · Zbl 1349.65476
[53] Rivière, B.; Wheeler, Mf; Girault, V., Improved energy estimates for interior penalty, constrained and discontinuous Galerkin methods for elliptic problems. Part I, Comput Geosci, 3, 3, 337-360 (1999) · Zbl 0951.65108
[54] Rivière, B.; Wheeler, Mf; Girault, V., A priori error estimates for finite element methods based on discontinuous approximation spaces for elliptic problems, SIAM J Numer Anal, 39, 3, 902-931 (2001) · Zbl 1010.65045
[55] Saichev, Ai; Zaslavsky, Gm, Fractional kinetic equations: solutions and applications, Chaos Interdiscip J Nonlinear Sci, 7, 4, 753-764 (1997) · Zbl 0933.37029
[56] Tadjeran, C.; Meerschaert, Mm, A second-order accurate numerical method for the two-dimensional fractional diffusion equation, J Comput Phys, 220, 2, 813-823 (2007) · Zbl 1113.65124
[57] Tarasov, Ve, Fractional dynamics: applications of fractional calculus to dynamics of particles, fields and media (2011), Berlin: Springer Science & Business Media, Berlin
[58] Wang, H.; Yang, D., Wellposedness of variable-coefficient conservative fractional elliptic differential equations, SIAM J Numer Anal, 51, 2, 1088-1107 (2013) · Zbl 1277.65059
[59] Wang, J.; Ye, X., A weak Galerkin finite element method for second-order elliptic problems, J Comput Appl Math, 241, 103-115 (2013) · Zbl 1261.65121
[60] Wheeler, Mf, An elliptic collocation-finite element method with interior penalties, SIAM J Numer Anal, 15, 1, 152-161 (1978) · Zbl 0384.65058
[61] Xu, Q.; Hesthaven, Js, Discontinuous Galerkin method for fractional convection-diffusion equations, SIAM J Numer Anal, 52, 1, 405-423 (2014) · Zbl 1297.26018
[62] Zaslavsky, Gm, Chaos, fractional kinetics, and anomalous transport, Phys Rep, 371, 6, 461-580 (2002) · Zbl 0999.82053
[63] Zaslavsky, G.; Edelman, M., Weak mixing and anomalous kinetics along filamented surfaces, Chaos Interdiscip J Nonlinear Sci, 11, 2, 295-305 (2001) · Zbl 1080.37584
[64] Zayernouri, M.; Karniadakis, Ge, Fractional Sturm-Liouville eigen-problems: theory and numerical approximation, J Comput Phys, 252, 495-517 (2013) · Zbl 1349.34095
[65] Zayernouri, M.; Karniadakis, Ge, Exponentially accurate spectral and spectral element methods for fractional ODEs, J Comput Phys, 257, Part A, 460-480 (2014) · Zbl 1349.65257
[66] Zhong Sun, Z.; Wu, X., A fully discrete difference scheme for a diffusion-wave system, Appl Numer Math, 56, 2, 193-209 (2006) · Zbl 1094.65083
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