  When solving the Poisson equation by the finite element method, we use one
degree of freedom for interpolation by the given Laplacian - the right hand
side function in the partial differential equation. The finite element solution
is the Galerkin projection in a smaller vector space. The idea is similar to
that of interpolating the boundary condition in the standard finite element
method. Due to the pointwise interpolation, our method yields a smaller system
of equations and a better condition number. The number of unknowns on each
element is reduced significantly from $(k^2+3k+2)/2$ to $3k$ for the $P_k$
($k\ge 3$) finite element. We construct 2D $P_2$ conforming and nonconforming,
and $P_k$ ($k\ge3$) conforming interpolated Galerkin finite elements on
triangular grids. This interpolated Galerkin finite element method is proved to
converge at the optimal order. Numerical tests and comparisons with the
standard finite elements are presented, verifying the theory and showing
advantages of the interpolated Galerkin finite element method.
