Overlapping Schwarz for Linear and Nonlinear Parabolic Problems
Martin J. Gander · Archive ouverte UNIGE (University of Geneva) · 1996
Introduction The basic ideas underlying waveform relaxation were first suggested in the late 19th century by Picard and Lindelof ([Lin94], [Lin93]). However much recent interest in waveform relaxation as a practical parallel method for the solution of stiff ordinary differential 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 field 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 finite time interval [0; T ]. This result can be found for