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Symmetric parareal algorithms for hamiltonian systems

Xiaoying Dai;  Claude Le Bris;  Frédéric Legoll;  Yvon Maday — 2013

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

The parareal in time algorithm allows for efficient parallel numerical simulations of time-dependent problems. It is based on a decomposition of the time interval into subintervals, and on a predictor-corrector strategy, where the propagations over each subinterval for the corrector stage are concurrently performed on the different processors that are available. In this article, we are concerned with the long time integration of Hamiltonian systems. Geometric, structure-preserving integrators are...

Adaptive finite element analysis based on perturbation arguments

Dai, Xiaoying;  He, Lianhua;  Zhou, Aihui — 2012

Applications of Mathematics 2012

We review some numerical analysis of an adaptive finite element method (AFEM) for a class of elliptic partial differential equations based on a perturbation argument. This argument makes use of the relationship between the general problem and a model problem, whose adaptive finite element analysis is existing, from which we get the convergence and the complexity of adaptive finite element methods for a nonsymmetric boundary value problem, an eigenvalue problem, a nonlinear boundary value problem...

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