An hp-version of the discontinuous Galerkin method for fractional integro-differential equations with weakly singular kernels

• Published in: BIT Vol. 64; no. 3
• Main Authors: Chen, Yanping, Chen, Zhenrong, Huang, Yunqing
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01.09.2024; Springer Nature B.V
• Subjects: Computational Mathematics and Numerical Analysis; Differential equations; Exact solutions; Galerkin method; Mathematical analysis; Mathematics; Mathematics and Statistics; Numeric Computing; Polynomials; Volterra integral equations; Fractional integro-differential equation; 65R20; version discontinuous Galerkin method; Weakly singular kernel; Exponential rate of convergence; Volterra integral equation; 45D05; 34A08
• ISSN: 0006-3835; 1572-9125
• DOI: 10.1007/s10543-024-01026-9

This paper suggests an hp -discontinuous Galerkin approach for the fractional integro-differential equations with weakly singular kernels. The key idea behind our method is to first convert the fractional integro-differential equations into the second kind of Volterra integral equations, and then solve the equivalent integral equations using the hp -discontinuous Galerkin method. We establish prior error bounds in the L 2 -norm that is entirely explicit about the local mesh sizes, local polynomial degrees, and local regularities of the exact solutions. The use of geometrically refined meshes and linearly increasing approximation orders demonstrates, in particular, that exponential convergence is achievable for solutions with endpoint singularities. Numerical results indicate the usefulness of the proposed method.