  By the simple finite element method, we study the symplectic, multisymplectic
structures and relevant preserving properties in some semi-linear elliptic
boundary value problem in one-dimensional and two-dimensional spaces
respectively. We find that with uniform mesh, the numerical schemes derived
from finite element method can keep a preserved symplectic structure in
one-dimensional case and a preserved multisymplectic structure in
two-dimentional case respectively. These results are in fact the intrinsic
reason that the numerical experiments indicate that such finite element schemes
are accurate in practice.
