A new Algorithm Based on Factorization for Heterogeneous Domain Decomposition

Often computational models are too expensive to be solved in the entire domain of simulation, and a cheaper model would suffice away from the main zone of interest. We present for the concrete example of an evolution problem of advection reaction diffusion type a heterogeneous domain decomposition algorithm which allows us to recover a solution that is very close to the solution of the fully viscous problem, but solves only an inviscid problem in parts of the domain. Our new algorithm is based on the factorization of the underlying differential operator, and we therefore call it factorization algorithm. We give a detailed error analysis, and show that we can obtain approximations in the viscous region which are much closer to the viscous solution in the entire domain of simulation than approximations obtained by other heterogeneous domain decomposition algorithms from the literature.Comment: 23 page

Cross-Points in Domain Decomposition Methods with a Finite Element Discretization

Non-overlapping domain decomposition methods necessarily have to exchange Dirichlet and Neumann traces at interfaces in order to be able to converge to the underlying mono-domain solution. Well known such non-overlapping methods are the Dirichlet-Neumann method, the FETI and Neumann-Neumann methods, and optimized Schwarz methods. For all these methods, cross-points in the domain decomposition configuration where more than two subdomains meet do not pose any problem at the continuous level, but care must be taken when the methods are discretized. We show in this paper two possible approaches for the consistent discretization of Neumann conditions at cross-points in a Finite Element setting

Is it possible to predict the far future before the near future is known accurately?

It has always been the dream of mankind to predict the future. If the future is governed by laws of physics, like in the case of the weather, one can try to make a model, solve the associated equations, and thus predict the future. However, to make accurate predictions can require extremely large amounts of computation. If we need seven days to compute a prediction for the weather tomorrow and the day after tomorrow, the prediction arrives too late and is thus not a prediction any more. Although it may seem improbable, with the advent of powerful computers with many parallel processors, it is possible to compute a prediction for tomorrow and the day after tomorrow simultaneously. We describe a mathematical algorithm which is designed to achieve this

Dirichlet-Neumann and Neumann-Neumann Waveform Relaxation for the Wave Equation

We present a Waveform Relaxation (WR) version of the Dirichlet-Neumann and Neumann-Neumann algorithms for the wave equation in space time. Each method is based on a non-overlapping spatial domain decomposition, and the iteration involves subdomain solves in space time with corresponding interface condition, followed by a correction step. Using a Laplace transform argument, for a particular relaxation parameter, we prove convergence of both algorithms in a finite number of steps for finite time intervals. The number of steps depends on the size of the subdomains and the time window length on which the algorithms are employed. We illustrate the performance of the algorithms with numerical results, and also show a comparison with classical and optimized Schwarz WR methods.Comment: 8 pages, 6 figures, presented in 22nd International conference on Domain Decomposition Methods, to appear in Domain Decomposition in Science and Engineering XXII, LNCSE, Springer-Verlag 201

A New Parareal Algorithm for Time-Periodic Problems with Discontinuous Inputs

The Parareal algorithm, which is related to multiple shooting, was introduced for solving evolution problems in a time-parallel manner. The algorithm was then extended to solve time-periodic problems. We are interested here in time-periodic systems which are forced with quickly-switching discontinuous inputs. Such situations occur, e.g., in power engineering when electric devices are excited with a pulse-width-modulated signal. In order to solve those problems numerically we consider a recently introduced modified Parareal method with reduced coarse dynamics. Its main idea is to use a low-frequency smooth input for the coarse problem, which can be obtained, e.g., from Fourier analysis. Based on this approach, we present and analyze a new Parareal algorithm for time-periodic problems with highly-oscillatory discontinuous sources. We illustrate the performance of the method via its application to the simulation of an induction machine