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In this paper, we revisit approximation properties of piecewise polynomialspaces, which contain more than ${\cal P}_{r-1}$ but not ${\cal P}_r$. Wedevelop more accurate upper and lower error bounds that are sharper than thoseused in literature. These new error bounds, especially the lower bounds areparticular useful to finite element methods. As an important application, weestablish sharp lower bounds of the discretization error for Laplace and$2m$-th order elliptic eigenvalue problems in various finite element spacesunder shape regular triangulations, and investigate the asymptotic convergencebehavior for large numerical eigenvalue approximations.

Sharp Error Bounds for Piecewise Polynomial Approximation: Revisit and Application to Elliptic PDE Eigenvalue Computation