Error analysis of time-stepping methods for integro-differential equations

For initial-value problems in ordinary diffential equations (ODEs) and parabolic partial dif- ferential equations (PDEs) time-stepping methods using the finite element method based on piecewise polynomials that are in general discontinuous at the points of the underlying temporal mesh (known as the discontinuous Galerkin (DG) method) has been studied exten- sively for the last twenty years and is now well understood. More recently, research has been focused on the error analysis (optimal order; a posteriori error estimation; computable error estimators) for the hp-version of the DG method: here, both the degrees of the piecewise polynomials and the time-step are allowed to vary from one subinterval to the next of the underlying mesh.

In many applications (heat condution; viscoelasticity; population growth) memory effects play a key role, and the resulting mathematical models are “non-local” (parabolic) PDEs; that is, they contain a memory term given by a Volterra integral operator that describes the history of the physical or biological process. Such equations are referred to as (parabolic) partial Volterra integro-differential equations (VIDEs). A typical example has the form ut-Δu = f(t,x)+ ∫_0^t▒〖k(t-s)Δu(s,·)ds〗, t> 0, x∈Ω, subject to appropriate initial and boundary conditions. Here, the kernel k(t- s) in the integral operator refects the history of the process, Ω ⊂ IRd(d = 1,2) denotes a (bounded spatial domain,andu=u(t,x) is the size of the unknown quantity (e.g. the temperature at time t and at the location x∈Ω.

The few papers on hp-DG methods for VIDEs that have appeared since 2006 deal with the analysis of the optimal order of convergence. However,the wider questions of adaptive time-stepping based on a posteriori error estimates and the derivation of computable error estimators remain to be answered. It is the aim of this research project to study these questions.