An Interpolated Galerkin Finite Element Method for the Poisson Equation

We develop a new approach to construct finite element methods solve the Poisson equation. The idea is use pointwise Laplacian as degree of freedom followed by interpolating solution at given right-hand side function in partial differential then Galerkin projection smaller vector space. This similar that boundary condition standard method. Our results system equations and better number. number unknowns on each reduced significantly from $$(k^2+3k+2)/2$$ 3k for $$P_k$$ ( $$k\ge 3$$ ) element. bivariate $$P_2$$ conforming nonconforming, interpolated elements triangular grids; prove their optimal order convergence; confirm our findings numerical tests.