On monotone iteration and Schwarz methods for nonlinear parabolic PDEs
Journal of Computational and Applied Mathematics
Domain decomposition methods in image denoising using Gaussian curvature
Journal of Computational and Applied Mathematics
Journal of Computational and Applied Mathematics
Optimized waveform relaxation methods for longitudinal partitioning of transmission lines
IEEE Transactions on Circuits and Systems Part I: Regular Papers - Special section on 2008 custom integrated circuits conference (CICC 2008)
Applied Numerical Mathematics
Optimization of Schwarz waveform relaxation over short time windows
Numerical Algorithms
A mathematical analysis of optimized waveform relaxation for a small RC circuit
Applied Numerical Mathematics
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Waveform relaxation algorithms for partial differential 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 refines 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 infinite 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 refinement. The algorithm is in the class of waveform relaxation algorithms based on overlapping multisplittings. Our analysis quantifies the empirical observation by Jeltsch and Pohl [SIAM J. Sci. Comput., 16 (1995), pp. 40--49] that the convergence rate of a multisplitting algorithm depends on the overlap.Numerical results are presented which support the convergence theory.