Optimal quadratic element on rectangular grids for $$H^1$$ problems

In this paper, a piecewise quadratic finite element method on rectangular grids for $$H^1$$ problems is presented. The proposed method can be viewed as a reduced rectangular Morley element. For the source problem, the convergence rate of this scheme is proved to be $$O(h^2)$$ in the energy norm on uniform grids over a convex domain. A lower bound of the $$L^2$$ -norm error is also proved, which makes the capacity of this scheme more clear. For the eigenvalue problem, the computed eigenvalues by this element are shown to be the lower bounds of the exact ones. Some numerical results are presented to verify the theoretical findings.