Error analysis of the L1 method on graded and uniform meshes for a fractional-derivative problem in two and three dimensions

Kopteva, Natalia

doi:10.1090/mcom/3410

Error analysis of the L1 method on graded and uniform meshes for a fractional-derivative problem in two and three dimensions
HTML articles powered by AMS MathViewer

by Natalia Kopteva;

Math. Comp. 88 (2019), 2135-2155

DOI: https://doi.org/10.1090/mcom/3410

Published electronically: January 23, 2019

HTML | PDF | Request permission

Abstract:

An initial-boundary value problem with a Caputo time derivative of fractional order $\alpha \in (0,1)$ is considered, solutions of which typically exhibit a singular behaviour at an initial time. For this problem, we give a simple framework for the analysis of the error of L1-type discretizations on graded and uniform temporal meshes in the $L_\infty$ and $L_2$ norms. This framework is employed in the analysis of both finite difference and finite element spatial discretiztions. Our theoretical findings are illustrated by numerical experiments.

References

Susanne C. Brenner and L. Ridgway Scott, The mathematical theory of finite element methods, 3rd ed., Texts in Applied Mathematics, vol. 15, Springer, New York, 2008. MR 2373954, DOI 10.1007/978-0-387-75934-0
Kai Diethelm, The analysis of fractional differential equations, Lecture Notes in Mathematics, vol. 2004, Springer-Verlag, Berlin, 2010. An application-oriented exposition using differential operators of Caputo type. MR 2680847, DOI 10.1007/978-3-642-14574-2
Lawrence C. Evans, Partial differential equations, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 1998. MR 1625845, DOI 10.1090/gsm/019
David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, Grundlehren der Mathematischen Wissenschaften, Vol. 224, Springer-Verlag, Berlin-New York, 1977. MR 473443
José Luis Gracia, Eugene O’Riordan, and Martin Stynes, Convergence in positive time for a finite difference method applied to a fractional convection-diffusion problem, Comput. Methods Appl. Math. 18 (2018), no. 1, 33–42. MR 3739302, DOI 10.1515/cmam-2017-0019
P. Grisvard, Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics, vol. 24, Pitman (Advanced Publishing Program), Boston, MA, 1985. MR 775683
Bangti Jin, Raytcho Lazarov, and Zhi Zhou, Error estimates for a semidiscrete finite element method for fractional order parabolic equations, SIAM J. Numer. Anal. 51 (2013), no. 1, 445–466. MR 3033018, DOI 10.1137/120873984
Bangti Jin, Raytcho Lazarov, and Zhi Zhou, Two fully discrete schemes for fractional diffusion and diffusion-wave equations with nonsmooth data, SIAM J. Sci. Comput. 38 (2016), no. 1, A146–A170. MR 3449907, DOI 10.1137/140979563
Bangti Jin, Raytcho Lazarov, and Zhi Zhou, An analysis of the L1 scheme for the subdiffusion equation with nonsmooth data, IMA J. Numer. Anal. 36 (2016), no. 1, 197–221. MR 3463438, DOI 10.1093/imanum/dru063
Bangti Jin, Raytcho Lazarov, Vidar Thomée, and Zhi Zhou, On nonnegativity preservation in finite element methods for subdiffusion equations, Math. Comp. 86 (2017), no. 307, 2239–2260. MR 3647957, DOI 10.1090/mcom/3167
Bangti Jin and Zhi Zhou, An analysis of Galerkin proper orthogonal decomposition for subdiffusion, ESAIM Math. Model. Numer. Anal. 51 (2017), no. 1, 89–113. MR 3601002, DOI 10.1051/m2an/2016017
Samir Karaa, Kassem Mustapha, and Amiya K. Pani, Optimal error analysis of a FEM for fractional diffusion problems by energy arguments, J. Sci. Comput. 74 (2018), no. 1, 519–535. MR 3742890, DOI 10.1007/s10915-017-0450-7
V. A. Kondrat′ev, Boundary value problems for elliptic equations in domains with conical or angular points, Trudy Moskov. Mat. Obšč. 16 (1967), 209–292 (Russian). MR 226187
Dmitriy Leykekhman and Boris Vexler, Finite element pointwise results on convex polyhedral domains, SIAM J. Numer. Anal. 54 (2016), no. 2, 561–587. MR 3470741, DOI 10.1137/15M1013912
Vladimir Maz’ya and Jürgen Rossmann, Elliptic equations in polyhedral domains, Mathematical Surveys and Monographs, vol. 162, American Mathematical Society, Providence, RI, 2010. MR 2641539, DOI 10.1090/surv/162
William McLean, Regularity of solutions to a time-fractional diffusion equation, ANZIAM J. 52 (2010), no. 2, 123–138. MR 2832607, DOI 10.1017/S1446181111000617
Ricardo H. Nochetto, Enrique Otárola, and Abner J. Salgado, A PDE approach to space-time fractional parabolic problems, SIAM J. Numer. Anal. 54 (2016), no. 2, 848–873. MR 3478958, DOI 10.1137/14096308X
Kenichi Sakamoto and Masahiro Yamamoto, Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, J. Math. Anal. Appl. 382 (2011), no. 1, 426–447. MR 2805524, DOI 10.1016/j.jmaa.2011.04.058
Alfred H. Schatz, A weak discrete maximum principle and stability of the finite element method in $L_{\infty }$ on plane polygonal domains. I, Math. Comp. 34 (1980), no. 149, 77–91. MR 551291, DOI 10.1090/S0025-5718-1980-0551291-3
A. H. Schatz and L. B. Wahlbin, On the quasi-optimality in $L_{\infty }$ of the $\dot H^{1}$-projection into finite element spaces, Math. Comp. 38 (1982), no. 157, 1–22. MR 637283, DOI 10.1090/S0025-5718-1982-0637283-6
Martin Stynes, Too much regularity may force too much uniqueness, Fract. Calc. Appl. Anal. 19 (2016), no. 6, 1554–1562. MR 3589365, DOI 10.1515/fca-2016-0080
Martin Stynes, Eugene O’Riordan, and José Luis Gracia, Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation, SIAM J. Numer. Anal. 55 (2017), no. 2, 1057–1079. MR 3639581, DOI 10.1137/16M1082329
E. A. Volkov, On differential properties of solutions of boundary value problems for the Laplace and Poisson equations on a rectangle, Trudy Mat. Inst. Steklov. 77 (1965), 89–112 (Russian). MR 192077
Jinchao Xu and Ludmil Zikatanov, A monotone finite element scheme for convection-diffusion equations, Math. Comp. 68 (1999), no. 228, 1429–1446. MR 1654022, DOI 10.1090/S0025-5718-99-01148-5