Spectral element method with geometric mesh for two-sided fractional differential equations
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Spectral element method with geometric mesh for two-sided fractional differential equations Zhiping Mao1 · Jie Shen2
Received: 15 February 2017 / Accepted: 9 August 2017 © Springer Science+Business Media, LLC 2017
Abstract Solutions of two-sided fractional differential equations (FDEs) usually exhibit singularities at the both endpoints, so it can not be well approximated by a usual polynomial based method. Furthermore, the singular behaviors are usually not known a priori, making it difficult to construct special spectral methods tailored for given singularities. We construct a spectral element approximation with geometric mesh, describe its efficient implementation, and derive corresponding error estimates. We also present ample numerical examples to validate our error analysis. Keywords Two-sided fractional differential equations · Singularity · Spectral element method · Geometric mesh · Error estimate · Exponential convergence Mathematics Subject Classification (2010) 65N35 · 65E05 · 65M70 · 41A05 · 41A10 · 41A25
Communicated by: Martin Stynes Jie Shen
[email protected] Zhiping Mao zhiping [email protected] 1
Division of Applied Mathematics, Brown University, 182 George St, Providence RI 02912, USA
2
Department of Mathematics, Purdue University, West Lafayette, IN 47907-1957, USA
Z. Mao and J. Shen
1 Introduction We consider numerical approximation for the two-sided fractional differential equations (FDEs): ρu − p1 −1 Dxα u(x) − p2 x D1α u(x) = f (x), x ∈ , u(±1) = 0,
(1.1)
where 1 < α < 2, ρ ≥ 0, p1 , p2 ≥ 0 and p1 + p2 = 0, f (x) is a given function, α α −1 Dx u(·) and x D1 u(·) are the left-sided and right-sided Riemann-Liouville (R-L) fractional derivative, respectively. FDEs provides a useful approach to describe transport dynamics in complex systems that are governed by anomalous diffusion and non-exponential relaxation patterns. In addition, the problem (1.1) arises when one discretizes in time parabolic equations with two-sided spatial fractional derivatives, for instance, fractional advection diffusion equations [5, 6, 14, 17, 18], fractional kinetic equation [9], fractional Fokker-Planck equation [22]. It is in general desirable to have high-order numerical methods when solving PDEs, including fractional PDEs. The convergence rate of numerical methods usually depends on the regularity of solutions in suitable functional spaces, e.g., usual Sobolev spaces for polynomial (local or global) based methods. However, it is now well-known that solutions of fractional PDEs usually do not have high regularities in the usual Sobolev spaces. Some high-order methods have been developed for FDEs such as (1.1), e.g., fourth order finite difference schemes [1, 27], spectral methods [13, 15] with the assumption that the solution of FDEs is sufficiently smooth in the usual Sobolev spaces, which does not hold in general. In some recent work, special treatments have been proposed to deal with the endpoint singularities for FDEs in some special cases of (1.1), such as • •
Left-sided FDEs: p1 = 0, p2 = 0
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