[Paper Review] A simple finite element method for the boundary value problem with a Riemann-Liouville derivative

Bangti Jin, Raytcho Lazarov+2 more|arXiv (Cornell University)|Feb 27, 2015

Fractional Differential Equations Solutions14 references3 citations

TL;DR

This paper proposes a novel finite element method for boundary value problems with a Riemann-Liouville fractional derivative of order $\alpha \in (3/2, 2)$, transforming the problem via a Riemann-Liouville integral to smooth the solution's singularity, enabling optimal $L^2$ convergence rates with standard finite elements. The method achieves optimal convergence and extends naturally to fractional Sturm-Liouville problems.

ABSTRACT

We consider a boundary value problem involving a Riemann-Liouville fractional derivative of order $α\in (3/2,2)$ on the unit interval $(0,1)$. The standard Galerkin finite element approximation converges slowly due to the presence of singularity term $x^{α-1}$ 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 $L^2(D)$ 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.

Motivation & Objective

• To address the slow convergence of standard Galerkin FEMs for fractional BVPs with Riemann-Liouville derivatives due to solution singularities.
• To develop a stable, efficient, and accurate finite element method that preserves optimal convergence rates despite the presence of $x^{\alpha-1}$ singularities.
• To extend the method to fractional Sturm-Liouville problems with Riemann-Liouville derivatives, enabling accurate eigenvalue approximations.
• To provide rigorous stability and error analysis for the transformed variational formulation and its finite element discretization.

Proposed method

• Transform the original fractional BVP into an equivalent second-order two-point boundary value problem with nonlocal low-order terms via the Riemann-Liouville integral operator ${}_0I_x^{2-\alpha}$.
• The transformation maps the singular term $x^{\alpha-1}$ into a smooth function $x$, enabling standard finite element approximation.
• Derive a stable variational formulation for the transformed problem using weighted Sobolev spaces and analyze its well-posedness.
• Apply conforming continuous piecewise linear and quadratic finite elements to the transformed problem, achieving optimal $L^2(D)$ error estimates.
• Reconstruct the original solution from the transformed solution using fractional differentiation, with stable numerical computation via expm1 function.
• Extend the method to the fractional Sturm-Liouville problem and derive convergence rates for eigenvalue approximations.

Experimental results

Research questions

• RQ1 Can a transformation-based finite element method achieve optimal $L^2$ convergence for fractional BVPs with Riemann-Liouville derivatives despite solution singularities?
• RQ2 Does transforming the problem via the Riemann-Liouville integral ${}_0I_x^{2-\alpha}$ effectively smooth the singularity $x^{\alpha-1}$ into a regular function?
• RQ3 Can the proposed method be extended to the fractional Sturm-Liouville problem with provable convergence for eigenvalue approximations?
• RQ4 What is the stability and regularity behavior of the variational formulation in the transformed variable?

Key findings

• The transformed problem admits a stable variational formulation with optimal regularity pickup, enabling optimal convergence rates for finite element approximations.
• Optimal $L^2(D)$ error estimates of order $O(h^2)$ are established for piecewise linear elements and $O(h^3)$ for quadratic elements.
• The method achieves improved convergence over standard Galerkin FEMs by resolving the singularity through transformation rather than explicit singularity subtraction.
• The approach is extendable to the fractional Sturm-Liouville problem, with convergence rates for eigenvalue approximations established.
• Numerical results confirm the theoretical error estimates and demonstrate the method’s efficiency and robustness across various test cases.
• Stable computation of the fractional derivative in the solution reconstruction is achieved using the expm1 function to avoid roundoff errors.