A simple finite element method for boundary value problems with a Riemann–Liouville derivative
KAUST Grant NumberKUS-C1-016-04
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Abstract© 2015 Elsevier B.V. All rights reserved. We consider a boundary value problem involving a Riemann-Liouville fractional derivative of order α∈(3/2,2) on the unit interval (0,1). The standard Galerkin finite element approximation converges slowly due to the presence of singularity term xα-<sup>1</sup> in the solution representation. In this work, we develop a simple technique, by transforming it into a second-order two-point boundary value problem with nonlocal low order terms, whose solution can reconstruct directly the solution to the original problem. The stability of the variational formulation, and the optimal regularity pickup of the solution are analyzed. A novel Galerkin finite element method with piecewise linear or quadratic finite elements is developed, and <sup>L2</sup>(D) error estimates are provided. The approach is then applied to the corresponding fractional Sturm-Liouville problem, and error estimates of the eigenvalue approximations are given. Extensive numerical results fully confirm our theoretical study.
CitationJin B, Lazarov R, Lu X, Zhou Z (2016) A simple finite element method for boundary value problems with a Riemann–Liouville derivative. Journal of Computational and Applied Mathematics 293: 94–111. Available: http://dx.doi.org/10.1016/j.cam.2015.02.058.
SponsorsThe authors are grateful to the anonymous referees for their insightful comments, which have led to improved presentation of the paper. The research of R. Lazarov was supported in parts by National Science Foundation Grant DMS-1016525 and also by Award No. KUS-C1-016-04, made by King Abdullah University of Science and Technology (KAUST). X. Lu is supported by National Natural Science Foundation of China Nos. 91230108 and 11471253. Z. Zhou was partially supported by National Science Foundation Grant DMS-1016525.