An hp-version of the C~0-continuous Petrov-Galerkin method for second-order Volterra integro-differential equations

摘要

We present an hp-version of the C~0-continuous Petrov-Galerkin method for linear second-order Volterra integro-differential equations. We combine the continuous and discontinuous Galerkin methodologies to obtain a natural discretization of the second-order derivative of arbitrary order. We derive a-priori error bounds in the I2- and H'-norm that are completely explicit in the local time steps, local approximation orders, and local regularity of the exact solution. Numerical experiments are provided to illustrate the theoretical results.