Semidiscrete finite element analysis of time fractional parabolic problems: a unified approach. (English) Zbl 1397.65188
The author considers time-fractional parabolic problems involving Caputo derivatives with respect to time of order \(\alpha\), \(0 < \alpha < 1\), and derives optimal error estimates for semidiscrete Galerkin finite element approximations for problems with smooth and nonsmooth initial data. An extension of this analysis to a multiterm time-fractional model is also discussed.
Reviewer: Murli Gupta (Washington, D. C.)
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 |
| 65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |
| 35R11 | Fractional partial differential equations |
Keywords:
time-fractional parabolic equation; multiterm fractional diffusion; semidiscrete finite element scheme; optimal error estimates; nonsmooth initial dataReferences:
| [1] | I. Babüska, The finite element method with Lagrangian multipliers, Numer. Math., 20 (1973), pp. 179–192. · Zbl 0258.65108 |
| [2] | D. Braess, Finite Elements: Theory, Fast Solvers, and Applications, in Elasticity Theory, 3rd ed., Cambridge University Press, Cambridge, UK, 2007. · Zbl 1180.65146 |
| [3] | J. H. Bramble, A. H. Schatz, V. Thomée, and L. B. Wahlbin, Some convergence estimates for semidiscrete Galerkin type approximations for parabolic equations, SIAM J. Numer. Anal., 14 (1977), pp. 218–241. · Zbl 0364.65084 |
| [4] | F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, 1st ed., Springer-Verlag, New York, 1991. · Zbl 0788.73002 |
| [5] | P. Chatzipantelidis, R. D. Lazarov, V. Thomée, and L. B. Wahlbin, Parabolic finite element equations in nonconvex polygonal domains, BIT, 46 (2016), pp. 113–143. · Zbl 1108.65097 |
| [6] | C. M. Chen, F. Liu, V. Anh, and I. Turner, Numerical methods for solving a two-dimensional variable-order anomalous sub-diffusion equation, Math. Comp., 81 (2012), pp. 345–366. · Zbl 1241.65077 |
| [7] | P. G. Ciarlet, The Finite Element Method for Elliptic Problems, Classics in Appl. Math. 40, SIAM, Philadelphia, 2002. · Zbl 0999.65129 |
| [8] | M. Crouzeix and P. Raviart, Conforming and nonconforming finite element methods for solving the stationary Stokes equations, RAIRO Anal. Numér., 7 (1973), pp. 33–76. · Zbl 0302.65087 |
| [9] | E. Cuesta, C. Lubich, and C. Palencia, Convolution quadrature time discretization of fractional diffusive-wave equations, Math. Comp., 75 (2006), pp. 673–696. · Zbl 1090.65147 |
| [10] | M. Cui, Compact alternating direction implicit method for two-dimensional time fractional diffusion equation, J. Comput. Phys., 231 (2012), pp. 262–2633. · Zbl 1242.65158 |
| [11] | R. Gorenflo, F. Mainardi, D. Moretti, and P. Paradisi, Time fractional diffusion: A discrete random walk approach, Nonlinear Dynam., 29 (2002), pp. 129–143. · Zbl 1009.82016 |
| [12] | D. Goswami and A. K. Pani, An alternate approach to optimal L2-error analysis of semidiscrete Galerkin methods for linear parabolic problems with nonsmooth initial data, Numer. Funct. Anal. Optim., 32 (2011), pp. 946–982. · Zbl 1243.65111 |
| [13] | B. Jin, R. Lazarov, J. Pascal, and Z. Zhou, Error analysis of semidiscrete finite element methods for inhomogeneous time-fractional diffusion, IMA J. Numer. Anal., 35 (2015), pp. 561–582. · Zbl 1321.65142 |
| [14] | B. Jin, R. Lazarov, and Z. Zhou, Error estimates for a semidiscrete finite element method for fractional order parabolic equations, SIAM J. Numer. Anal., 51 (2013), pp. 445–466. · Zbl 1268.65126 |
| [15] | B. Jin, R. Lazarov, and Z. Zhou, The Galerkin finite element method for a multi-term time-fractional diffusion equation, J. Comput. Phys., 281 (2015), pp. 825–843. · Zbl 1352.65350 |
| [16] | B. Jin, R. Lazarov, and Z. Zhou, Two fully discrete schemes for fractional diffusion and diffusion-wave equations with nonsmooth data, SIAM J. Sci. Comput., 38 (2016), pp. A164–A170. · Zbl 1381.65082 |
| [17] | C. Johnson and V. Thomée, Error estimates for some mixed finite element methods for parabolic type problems, RAIRO Anal. Numér., 14 (1981), pp. 41–78. · Zbl 0476.65074 |
| [18] | S. Karaa, K. Mustapha, and A. K. Pani, Finite volume element method for two-dimensional fractional subdiffusion problems, IMA J. Numer. Anal., 37 (2017), pp. 945–964. · Zbl 1433.65214 |
| [19] | S. Karaa, K. Mustapha, and A. K. Pani, Optimal error analysis of a FEM for fractional diffusion problems by energy arguments, J. Sci. Comput., 74 (2018), pp. 519–535. · Zbl 1398.65228 |
| [20] | S. Karaa and A. K. Pani, Error analysis of a FVEM for fractional order evolution equations with nonsmooth initial data, ESAIM Math. Model. Numer. Anal., to appear. · Zbl 1404.65114 |
| [21] | A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006. · Zbl 1092.45003 |
| [22] | K. N. Le, W. McLean, and B. Lamichhane, Finite element approximation of a time-fractional diffusion problem for a domain with a re-entrant corner, ANZIAM J., 59 (2017), pp. 61–82. · Zbl 1404.65273 |
| [23] | K. N. Le, W. McLean, and K. Mustapha, Numerical solution of the time-fractional Fokker–Planck equation with general forcing, SIAM J. Numer. Anal., 54 (2016), pp. 1763–1784. · Zbl 1404.65122 |
| [24] | X. Li, X. Yang, and Y. Zhang, Error estimates of mixed finite element methods for time-fractional Navier-Stokes equations, J. Sci. Comput., 70 (2017), pp. 500–515. · Zbl 1383.65107 |
| [25] | W. Mclean, Regularity of solutions to a time-fractional diffusion equation, ANZIAM J., 52 (2010), pp. 123–138. · Zbl 1228.35266 |
| [26] | W. McLean, Fast summation by interval clustering for an evolution equation with memory, SIAM J. Sci. Comput., 34 (2012), pp. A3039–A3056. · Zbl 1276.65059 |
| [27] | W. McLean and V. Thomée, Numerical solution via Laplace transforms of a fractional order evolution equation, J. Integral Equations Appl., 22 (2010), pp. 57–94. · Zbl 1195.65122 |
| [28] | W. McLean and V. Thomée, Maximum-norm error analysis of a numerical solution via Laplace transformation and quadrature of a fractional order evolution equation, IMA J. Numer. Anal., 30 (2010), pp. 208–230. · Zbl 1416.65381 |
| [29] | E. W. Montroll and G. H. Weiss, Random walks on lattices. II, J. Math. Phys., 6 (1965), pp. 167–181. · Zbl 1342.60067 |
| [30] | K. Mustapha, FEM for time-fractional diffusion equations, novel optimal error analyses, Math. Comp., 87 (2018), pp. 2259–2272. · Zbl 1394.65086 |
| [31] | K. Mustapha and W. McLean, Piecewise-linear, discontinuous Galerkin method for a fractional diffusion equation, Numer. Algorithms, 56 (2011), pp. 159–184. · Zbl 1211.65127 |
| [32] | K. Mustapha and D. Schötzau, Well-posedness of \(hp\)-version discontinuous Galerkin methods for fractional diffusion wave equations, IMA J. Numer. Anal., 34 (2014), pp. 1226–1246. |
| [33] | J. A. Nitsche, Über ein Variationsprinzip zur Lösung yon Dirichlet-Problemen bei Verwendung von Teilrädumen, die keinen Randbedingungen unterworfen sind, Abh. Math. Sem. Univ. Hamburg, 36 (1971), pp. 9–15. · Zbl 0229.65079 |
| [34] | P. Raviart and J. A. Thomas, Mixed finite element method for second order elliptic problems, in Mathematical Aspects of the Finite Element Method, Lecture Notes in Math. 606, I. Galligani and E. Magenes, eds., Springer-Verlag, Berlin, 1977. · Zbl 0362.65089 |
| [35] | K. Sakamoto and M. Yamamoto, Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, J. Math. Anal. Appl., 382 (2011), pp. 426–447. · Zbl 1219.35367 |
| [36] | V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, 2nd ed., Springer-Verlag, Berlin, 1997. · Zbl 0884.65097 |
| [37] | V. Thomée, Negative norm estimates and superconvergence in Galerkin methods for parabolic problems, Math. Comp., 34 (1980), pp. 93–113. · Zbl 0454.65077 |
| [38] | Y. N. Zhang and Z. Z. Sun, Alternating direction implicit schemes for the two-dimensional fractional sub-diffusion equation, J. Comput. Phys., 230 (2011), pp. 8713–8728. · Zbl 1242.65174 |
| [39] | Y. Zhao, P. Chen, W. Bu, X. Liu, and Y. Tang, Two mixed finite element methods for time-fractional diffusion equations, J. Sci. Comput., 70 (2017), pp. 407–428. · Zbl 1360.65245 |
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