Symplectic And Multisymplectic Structure-Preserving In Mixed Finite Element Method

We all know that both finite element method(FEM), symplectic algorithm and multi-symplectic algorithm are powerful tools to solve partial differential equations numerically. This dissertation is devoted to studying some relations between them.The most important property of Hamiltonian systems is the Poincare and Liuville's conservation law of phase areas, i.e., the phaseflow is a one-parameter symplectic transformation. In numerically solving these equations, we hope that the numerical schemes can hold this property and the corresponding numerical methods are called as symplectic methods.In the discrete mechanics or field theories, discrete variation problems play an important role, particularly, for the discrete Hamiltonian formalism. Recently, the difference discrete variational principle(DDVP) approach has been proposed in the both difference discrete Lagrangian and Hamiltonian formalisms that relate each other by discrete Leg-endre transformation or its covariant form. Similarly, whether the mixed finite element schemes also have some hidden symplectic and multisymplectic structures is the important problem which has been discussed in this paper.In this paper, after we propose the mixed finite element discrete variational approach for the semilinear elliptic equation in one-dimensional and two-dimensional space, we get the mixed finite element discrete version of equation and symplectic/multisymplectic structure preserving properties, basing on the Hamilton's principle and nilpotency of exterior differential operator. And we have also proved the necessary and sufficient condition for the mixed finite element symplictic/multisymplectic conservation laws is directly related to the kernel of the relevant first EL(Euler-Lagrange) cohomology rather than the solution space of the equation of motions.