New expansions of numerical eigenvalues for $-\Delta u=\lambda \rho u$ by nonconforming elements

The paper explores new expansions of the eigenvalues for $-\Delta u=\lambda \rho u$ in $S$ with Dirichlet boundary conditions by the bilinear element (denoted $Q_1$) and three nonconforming elements, the rotated bilinear element (denoted $Q_1^{rot}$), the extension of $Q_1^{rot}$ (denoted $EQ_1^{rot}$) and Wilson's elements. The expansions indicate that $Q_1$ and $Q_1^{rot}$ provide upper bounds of the eigenvalues, and that $EQ_1^{rot}$ and Wilson's elements provide lower bounds of the eigenvalues. By extrapolation, the $O(h^4)$ convergence rate can be obtained, where $h$ is the maximal boundary length of uniform rectangles. Numerical experiments are carried out to verify the theoretical analysis made.