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

Abstract

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.