Space-Time Continuous Analysis of Waveform Relaxation for the Heat Equation

Waveform relaxation algorithms for partial diierential equations (PDEs) are traditionally obtained by discretizing the PDE in space and then splitting the discrete operator using matrix splittings. For the semidiscrete heat equation one can show linear convergence on unbounded time intervals and superlinear convergence on bounded time intervals by this approach. However the bounds depend in general on the mesh parameter and convergence rates deteriorate as one reenes the mesh. Motivated by the original development of waveform relaxation in circuit simulation, where the circuits are split in the physical domain into subcircuits, we split the PDE by using overlapping domain decomposition. We prove linear convergence of the algorithm in the continuous case on an innnite time interval, at a rate depending on the size of the overlap. This result remains valid after discretization in space and the convergence rates are robust with respect to mesh reenement. The algorithm is in the class of waveform relaxation algorithms based on overlapping multi-splittings. Our analysis quantiies the empirical observation by Jeltsch and Pohl SISC, 16 no. 1 (1995)] that the convergence rate of a multi-splitting algorithm depends on the overlap. Numerical results are presented which support the convergence theory.