Numerical analysis of Volterra integral equations with highly oscillatory kernels

Integral equations arising in the mathematical modelling of retarded potentials or elastodynamic models in fracture interaction often contain kernels that are highly oscillatory. The efficient and accurate computational solution of the resulting Volterra-type integral equations (reflecting memory effects in the model) is very difficult, and so far both the theory and the design of suitable numerical schemes are essentially open.

In this proposal we address these challenges, exploiting recent advances in the approximation of highly oscillatory integrals and in discontinuous Galerkin methods for Volterra integral equations of the first and second kind. The derivation of numerical schemes hinges on new results on the nature of solutions to such highly oscillatory Volterra integral equations, since highly oscillatory kernels may or may not lead to highly oscillatory solutions.

We also use the above insight and results to study the computational solution of singularly perturbed Volterra equations, and (ill-posed) first-kind Volterra integral equations with noisy data.