SciCADE 2013
International Conference on Scientific Computation and Differential Equations
September 16-20, 2013, Valladolid (Spain)

Invited Talk

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On the derivation of energy-preserving $H^1$-Galerkin schemes for Hamiltonian partial differential equations

Y. Miyatake and T. Matsuo

Abstract
In this talk, we propose an energy-preserving finite element method for Hamiltonian PDEs. As is well known, energy-preserving integrators often give qualitatively nice solutions, and on the other hand, $H^1$-formulations are generally preferred in the finite element discretizations from the practical point of view. A difficulty arises in the attempt of combining these two concepts, when the structure of the equation is complicated. For example, for the equation which has higher order derivatives, we usually consider a mixed formulation so that the equation can be formulated in $H^1$ space. However, this generally destroys the structure of the equation. Our method successfully solves such a difficulty, and is applicable to not only a wide variety of Hamiltonian PDEs but also dissipative PDEs. We can adopt the discrete gradient method for finding fully discrete schemes. The new method can be combined also with symplectic time discretizations.

Organized by         Universidad de Valladolid     IMUVA