Exponential Convergence of hp-discontinuous Galerkin Method for Nonlinear Caputo Fractional Differential Equations

We present an hp-discontinuous Galerkin method for solving nonlinear fractional differential equations involving Caputo-type fractional derivative. The main idea behind our approach is to first transform the fractional differential equations into nonlinear Volterra or Fredholm integral equations, and then the hp-discontinuous Galerkin method is used to solve the equivalent integral equations. We derive a-priori error bounds in the L-2-norm that are totally explicit with respect to the local mesh sizes, the local polynomial degrees, and the local regularities of the exact solutions. In particular, we prove that exponential convergence can be achieved for solutions with endpoint singularities by using geometrically refined meshes and linearly increasing approximation orders. The theoretical results are confirmed by a series of numerical experiments.