Fast Nyström Methods for Parabolic Boundary Integral Equations

Abstract Time dependence in parabolic boundary integral operators appears in form of an integral over the previous time evolution of the problem. The kernels are sin-gular only at the current time and get increasingly smooth for contributions that are further back in time. The thermal layer potentials can be regarded as generalized Abel operators where the kernel is a parameter dependent surface integral operator. This special form implies that discretization methods and fast evaluation methods must be signicantly changed from the familiar elliptic case. After a brief review of recent developments in the area we discuss the different options to discretize Abel integral operators in time. These methods are combined with standard surface quadrature rules to obtain a Nystr¤om method for parabolic integral equations. The method is explicit and we will show how a version of the fast multipole method in space and time can be used to evaluate the time stepping scheme efciently.