Abstract
We derive optimal \(L^2\)-error estimates for semilinear time-fractional subdiffusion problems involving Caputo derivatives in time of order \(\alpha \in (0,1)\), for cases with smooth and nonsmooth initial data. A general framework is introduced allowing a unified error analysis of Galerkin type space approximation methods. The analysis is based on a semigroup type approach and exploits the properties of the inverse of the associated elliptic operator. Completely discrete schemes are analyzed in the same framework using a backward Euler convolution quadrature method in time. Numerical examples including conforming, nonconforming and mixed finite element methods are presented to illustrate the theoretical results.
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References
Al-Maskari, M., Karaa, S.: Numerical approximation of semilinear subdiffusion equations with nonsmooth initial data. SIAM J. Numer. Anal. 57, 1524–1544 (2019)
Arnold, D.N., Brezzi, F., Cockburn, B., Marini, L.D.: Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39, 1749–1779 (2002)
Bramble, J.H., Schatz, A.H., Thomée, V., Wahlbin, L.B.: Some convergence estimates for semidiscrete Galerkin type approximations for parabolic equations. SIAM J. Numer. Anal. 14, 218–241 (1977)
Crouzeix, M., Raviart, P.: Conforming and nonconforming finite element methods for solving the stationary Stokes equations. RAIRO Anal. Numér. 7, 33–76 (1973)
Cuesta, E., Lubich, C., Palencia, C.: Convolution quadrature time discretization of fractional diffusion-wave equations. Math. Comput. 75, 673–696 (2006)
Chen, C., Thomée, V., Wahlbin, L.B.: Finite element approximation of a parabolic integro-differential equation with a weakly singular kernel. Math. Comput. 58, 587–602 (1992)
Chen, Z.: Expanded mixed finite element methods for linear second-order elliptic problems, I. RAIRO Modél. Math. Anal. Numér. 32, 479–499 (1998)
Cockburn, B., Mustapha, K.: A hybridizable discontinuous Galerkin method for fractional diffusion problems. Numer. Math. 130, 293–314 (2015)
Dixon, J., McKee, S.: Weakly singular discrete Gronwall inequalities. Z. Angew. Math. Mech. 66, 535–544 (1986)
Hecht, F., Pironneau, O., Le Hyaric, A.: www.freefem.org/ff++
Jin, B., Lazarov, R., Zhou, Z.: Two fully discrete schemes for fractional diffusion and diffusion-wave equations with nonsmooth data. SIAM J. Sci. Comput. 38, A146–A170 (2016)
Jin, B., Lazarov, R., Liu, Y., Zhou, Z.: The Galerkin finite element method for a multi-term time-fractional diffusion equation. J. Comput. Phys. 281, 825–843 (2015)
Jin, B., Lazarov, R., Zhou, Z.: Numerical methods for time-fractional evolution equations with nonsmooth data: a concise overview. Comput. Methods Appl. Mech. Eng. 346, 332–358 (2019)
Jin, B., Li, B., Zhou, Z.: Numerical Analysis of nonlinear subdiffusion equations. SIAM J. Numer. Anal. 56, 1–23 (2018)
Johnson, C., Larsson, S., Thomée, V., Wahlbin, L.B.: Error estimates for spatially discrete approximations of semilinear parabolic equations with nonsmooth initial data. Math. Comput. 49, 331–357 (1987)
Johnson, C., Thomée, V.: Error estimates for some mixed finite element methods for parabolic type problems. RAIRO Anal. Numér. 14, 41–78 (1981)
Karaa, S.: Semidiscrete finite element analysis of time fractional parabolic problems: a unified approach. SIAM J. Numer. Anal. 56, 1673–1692 (2018)
Karaa, S., Pani, A.K.: Mixed FEM for time-fractional diffusion problems with time-dependent coefficients. J Sci Comput. (2020). https://doi.org/10.1007/s10915-020-01236-7
Karaa, S., Pani, A.K.: Error analysis of a FVEM for fractional order evolution equations with nonsmooth initial data. ESAIM Math. Model. Numer. Anal. 52, 773–801 (2018)
Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)
Li, D., Liao, H., Sun, W., Wang, J., Zhang, J.: Analysis of \(L^1\)-Galerkin FEMs for time-fractional nonlinear parabolic problems. Commun. Comput. Phys. 24, 86–103 (2017)
Li, D., Wang, J., Zhang, J.: Unconditionally convergent \(L^1\)-Galerkin FEMs for nonlinear time-fractional Schrödinger equations. SIAM J. Sci. Comput. 39, A3067–A3088 (2017)
Li, X., Yang, X., Zhang, Y.: Error estimates of mixed finite element methods for time-fractional Navier–Stokes equations. J. Sci. Comput. 70, 500–515 (2017)
Lubich, C.: Discretized fractional calculus. SIAM J. Math. Anal. 17, 704–719 (1986)
Lubich, C.: Convolution quadrature and discretized operational calculus-I. Numer. Math. 52, 129–145 (1988)
Lubich, C., Sloan, I.H., Thomée, V.: Nonsmooth data error estimates for approximations of an evolution equation with a positive-type memory term. Math. Comput. 65, 1–17 (1996)
McLean, W., Thomée, V.: Numerical solution via Laplace transforms of a fractional order evolution equation. J. Integral Equ. Appl. 22, 57–94 (2010)
McLean, W., Thomée, V.: Maximum-norm error analysis of a numerical solution via Laplace transformation and quadrature of a fractional order evolution equation. IMA J. Numer. Anal. 30, 208–230 (2010)
Mustapha, K., Mustapha, H.: A second-order accurate numerical method for a semilinear integro-differential equation with a weakly singular kernel. IMA J. Numer. Anal. 30, 555–578 (2010)
Nitsche, J.A.: Über ein Variationsprinzip zur Lösung yon Dirichlet-Problemen bei Verwendung von Teilrädumen, die keinen Randbedingungen unterworfen sind. Abh. Math. Sem. Univ. Hamburg 36, 9–15 (1971)
Raviart, P., Thomas, J.A.: Mixed finite element method for second order elliptic problems. In: Galligani, I., Magenes, E. (eds.) Mathematical Aspects of the Finite Element Method. Lecture Notes in Mathematics, vol. 606. Springer, Berlin (1977)
Thomée, V.: Galerkin finite element methods for parabolic problems. Springer, Berlin (1997)
Zhao, Y., Chen, P., Bu, W., Liu, X., Tang, Y.: Two mixed finite element methods for time-fractional diffusion equations. J. Sci. Comput. 70, 407–428 (2017)
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This research was supported by the Research Council of Oman Grant ORG/CBS/15/001.
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Karaa, S. Galerkin Type Methods for Semilinear Time-Fractional Diffusion Problems. J Sci Comput 83, 46 (2020). https://doi.org/10.1007/s10915-020-01230-z
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DOI: https://doi.org/10.1007/s10915-020-01230-z
Keywords
- Semilinear fractional diffusion
- Galerkin method
- Nonconforming FE method
- Mixed FE method
- Convolution quadrature
- Error estimate