Hp-Discontinuous Galerkin Time-Stepping for Volterra Integrodifferential Equations

Abstract

We present an $hp$-error analysis of the discontinuous Galerkin time-stepping method for Volterra integrodifferential equations with weakly singular kernels. We derive new error bounds that are explicit in the time-steps, the degrees of the approximating polynomials, and the regularity properties of the exact solution. It is then shown that start-up singularities can be resolved at exponential rates of convergence by using geometrically graded time-steps. Our theoretical results are confirmed in a series of numerical tests.