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We study the numerical scheme of elliptic equations by the finite element method. With the special finite element domain, we can find that the scheme can keep a preserved symplectic structure in one-dimensional case and a preserved multisymplectic structure in two-dimensional case. Then we consider the discrete variational principle with the finite element method in the corresponding Lagrangian formalism...
Kauffman bracket polynomials of the so-called generalized tree-like links are studied. An algorithm of Witten type invariants, which was defined by Blanchet and Habegger et al. of more general 3-manifolds is given.
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