1Overlapping Schwarz for Linear and Nonlinear Parabolic Problems

The basic ideas underlying waveform relaxation were rst suggested in the late 19th century by Picard and Lindell of ((Lin94], Lin93]). However much recent interest in waveform relaxation as a practical parallel method for the solution of stii ordinary diierential equations (ODE's) has been generated after the publication of a paper by Lelarasmee and coworkers LRSV82] in the VLSI literature, and the paper by O'Leary and White OW85] which introduced multi-splittings of matrices for the solution of linear systems of equations. Recent work in this eld includes papers by Miekkala and Nevanlinna MN87a], MN87b], Nevanlinna Nev89a], Nev89b] Bellen and Zennaro BZ93] and Jeltsch and Pohl JP95]. The standard convergence result for a system of nonlinear ODE's needs the assumption that the splitting function is Lipschitz continuous in both arguments. It states superlinear convergence on any nite time interval 0; T]. This result can be found for example in Bjjrhus Bjj95a]. For a linear system of ODE's which is asymptotically stable Miekkala and Nevanlinna show in MN87a] the existence of splittings such that the waveform relaxation algorithm converges linearly on the innnite time interval 0; 1). Jeltsch and Pohl JP95] extend the work of MN87a] to prove superlinear convergence of certain overlapping splittings on bounded time intervals. They also extend the results on unbounded time intervals to overlapping splittings for a certain class of problems. However in all the results mentioned the constants in general depend badly on x if the linear ODE arises from a partial diierential equation (PDE) which is discretized in space. Motivated by the work of Bjjrhus Bjj95b], we show how one can use overlapping domain decomposition to obtain a waveform relaxation algorithm for the semi-discrete heat equation which converges at a rate independent of the mesh parameter x. The details of the analysis can be found in GS96].