Sharp Error Bounds for Piecewise Polynomial Approximation: Revisit and Application to Elliptic PDE Eigenvalue Computation
Provides refined theoretical error bounds for finite element methods, benefiting researchers working on numerical analysis of elliptic eigenvalue problems.
The paper develops sharper upper and lower error bounds for piecewise polynomial spaces and applies them to establish sharp lower bounds for discretization errors in elliptic eigenvalue problems, improving upon existing bounds in the literature.
In this paper, we revisit approximation properties of piecewise polynomial spaces, which contain more than ${\cal P}_{r-1}$ but not ${\cal P}_r$. We develop more accurate upper and lower error bounds that are sharper than those used in literature. These new error bounds, especially the lower bounds are particular useful to finite element methods. As an important application, we establish sharp lower bounds of the discretization error for Laplace and $2m$-th order elliptic eigenvalue problems in various finite element spaces under shape regular triangulations, and investigate the asymptotic convergence behavior for large numerical eigenvalue approximations.