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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.