An hp-version of the discontinuous Galerkin time-stepping method for Volterra integral equations with weakly singular kernels

摘要

We develop and analyze an hp-version of the discontinuous Galerkin time-stepping method for linear Volterra integral equations with weakly singular kernels. We derive a priori error bound in the L~2-norm that is fully explicit in the local time steps, the local approximation orders, and the local regularity of the exact solutions. For solutions with singular behavior near t = 0 caused by the weakly singular kernels, we prove optimal algebraic convergence rates for the h-version of the discontinuous Galerkin approximations on graded meshes. Moreover, we show that exponential rates of convergence can be achieved for solutions with start-up singularities by using geometrically refined time steps and linearly increasing approximation orders. Numerical experiments are presented to illustrate the theoretical results.