The Finite Element Discretization Of The Variational Complex

A variational complex is a theoretical tool for the rigorous study of Lagrangian systems. Using an exact variational complex can solve many problems about the variational calculus.This paper gets the discrete version of the continuous horizontal complex via the finite element method, the finite element form of the vertical complex by virtue of the cochain, the vertical functional complex produced by the equivalence relation of the vertical forms through a coboundary operator. On the basis of the findings, the paper draws the finite element variational complex, then proves its exactness property by means of the construction of a sequence of homotopy operators.Besides, Chapter Three gets the Euler-Lagrange cohomology in the finite element discrete Lagrangian mechanics via the difference discrete variational principle.