Sylleptic And Multisyllabic Structure-Preserving For 3-D Mixed Finite Element Method

There are generally two common discrete ways for differential equations, which are finite element method and finite difference method. This article is devoted to studying the relations between them.At first we can get the Lagrangian function from the elliptic equation, then replace differential coefficient by difference and get the discrete Lagrangian function. By the way, the Lagrange function can also be representable in Hamilton's system. More, we go on discretion it by finite element method. Basing on the Hamilton's principle and nilpotent of exterior differential operator, the finite element discrete version of equation and symplectic/multisymplectic structure preserving properties are gained.Hamiltonian system is one of important system in dynamics because all earthy physical course without any loss of energy must be classified in Hamiltonian system. The discrete algorithm to solving Hamiltonian is just called sylleptic algorithm. One result of this article is the symplectic/multisymplectic structure preserving properties, and it sure that our arithmetic is sylleptic algorithm.We can discrete a differential equation by finite element method. But when discrete system of equations, we need an B.-B. condition in addition when using mixed finite element method. After that we may gain the result from disctete Lagrangian functional.About the structure preserving properties, we prove the necessary and sufficient condition for symplectic/multisymplectic conservation laws is directly related to the kernel of the relevant first Euler-Lagrange cohomology rather than the solution space of the equation of motions.