Waveform Relaxation has been applied primarily to solve ordinary differential equations, possibly obtained by semi-discretization from a time-dependent PDE. The convergence theory of the method for that type of problem is nowadays well understood. Recently, however, the method was applied to delay differential equations, or, more generally, functional-differential equations. Such equations arise for example in population dynamics, and in the study of nonlinear materials with memory. Earlier studies concentrated on simple Jacobi- and Gauss-Seidel type iterations. In this seminar we will concentrate on the multigrid acceleration of those methods.
First the type of equation that is consider will be defined. It will be shown that delay partial differential equations exhibit quite different stability characteristics than classical partial differential equations. Then, the application of waveform relaxation and its multigrid acceleration will be illustrated by means of a number of examples. Finally, the convergence of the method will be studied by using a Fourier analysis technique, which takes the interaction of error smoothing and coarse grid correction into account. Numerical results will be supplied to illustrate the theory.