A Finite Element Method with Singularity Reconstruction for Fractional Boundary Value Problems

Jin, Bangti; Zhou, Zhi

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 49 (2015) no. 5, pp. 1261-1283.

We consider a two-point boundary value problem involving a Riemann−Liouville fractional derivative of order $\alpha \in (1,2)$ in the leading term on the unit interval $(0,1)$. The standard Galerkin finite element method can only give a low-order convergence even if the source term is very smooth due to the presence of the singularity term $x^{\alpha -1}$ in the solution representation. In order to enhance the convergence, we develop a simple singularity reconstruction strategy by splitting the solution into a singular part and a regular part, where the former captures explicitly the singularity. We derive a new variational formulation for the regular part, and show that the Galerkin approximation of the regular part can achieve a better convergence order in the $L^2(0,1)$, $H^{\alpha /2}(0,1)$ and $L^{\infty }(0,1)$-norms than the standard Galerkin approach, with a convergence rate for the recovered singularity strength identical with the $L^2(0,1)$ error estimate. The reconstruction approach is very flexible in handling explicit singularity, and it is further extended to the case of a Neumann type boundary condition on the left end point, which involves a strong singularity $x^{\alpha -2}$. Extensive numerical results confirm the theoretical study and efficiency of the proposed approach.

R.A. Adams and J.J.F. Fournier, Sobolev Spaces, 2nd edition. Elsevier/Academic Press, Amsterdam (2003). | Zbl

Z. Bai and H. Lu, Positive solutions for boundary value problem of nonlinear fractional differential equation. J. Math. Anal. Appl. 311 (2005) 495–505,. | DOI | Zbl

D.A. Benson, S.W. Wheatcraft and M.M. Meerschaert, The fractional-order governing equation of Lévy motion. Water Resour. Res. 36 (2000) 1413–1424. | DOI

Z. Cai and S. Kim, A finite element method using singular functions for the Poisson equation: corner singularities. SIAM J. Numer. Anal. 39 (2001) 286–299. | DOI | Zbl

D. Del-Castillo-Negrete, B.A. Carreras and V.E. Lynch, Front dynamics in reaction-diffusion systems with Levy flights. Phys. Rev. Lett. 91 (2003) 018302. | DOI

D. Del-Castillo-Negrete, B.A. Carreras and V.E. Lynch, Nondiffusive transport in plasma turbulence: a fractional diffusion approach. Phys. Rev. Lett. 94 (2005) 065003. | DOI

W.H. Deng and J.S. Hesthaven, Local discontinuous Galerkin methods for fractional diffusion equations. ESAIM: M2AN 47 (2013) 1845–1864. | DOI | Numdam | Zbl

J. Douglas Jr., and T. Dupont, Galerkin approximations for the two point boundary problem using continuous, piecewise polynomial spaces. Numer. Math. 22 (1974) 99–109. | DOI | Zbl

A. Ern and J.-L. Guermond, Theory and Practice of Finite Elements. Springer-Verlag, New York (2004). | Zbl

V.J. Ervin and J.P. Roop, Variational formulation for the stationary fractional advection dispersion equation. Numer. Methods Partial Differ. Eqs. 22 (2006) 558–576. | DOI | Zbl

B. Jin, R. Lazarov, J. Pasciak and W. Rundell, Variational formulation of problems involving fractional order differential operators. To appear in Math. Comput. (2015).

B. Jin, R. Lazarov, J. Pasciak and Z. Zhou, Error analysis of a finite element method for the space-fractional parabolic equation. SIAM J. Numer. Anal. 52 (2014) 2272–2294. | DOI | Zbl

A. Kilbas, H. Srivastava and J. Trujillo, Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006). | Zbl

R. Metzler and J. Klafter, The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339 (2000) 1–77. | DOI | Zbl

S.G. Samko, A.A. Kilbas and O.I. Marichev. Fractional Integrals and Derivatives. Gordon and Breach, Yverdon (1993). | Zbl

A.H. Schatz, An observation concerning Ritz-Galerkin methods with indefinite bilinear forms. Math. Comput. 28 (1974) 959–962. | DOI | Zbl

E. Sousa, Finite difference approximations for a fractional advection diffusion problem. J. Comput. Phys. 228 (2009) 4038–4054. | DOI | Zbl

C. Tadjeran and M.M. Meerschaert, A second-order accurate numerical method for the two-dimensional fractional diffusion equation. J. Comput. Phys. 220 (2007) 813–823. | DOI | Zbl

W. Tian, H. Zhou and W. Deng, A class of second-order finite difference approximations for solving space fractional diffusion equations. To appear in Math. Comput. (2015).

H. Wang and D. Yang, Wellposedness of variable-coefficient conservative fractional elliptic differential equations. SIAM J. Numer. Anal. 51 (2013) 1088–1107. | DOI | Zbl

K. Yoshida, Functional Analysis, 6th edition. Springer-Verlag, Berlin (1980).