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

TL;DR

This paper refines error bounds for piecewise polynomial approximation and applies these results to improve understanding of eigenvalue errors in finite element methods for elliptic PDEs.

Contribution

It develops sharper upper and lower error bounds for piecewise polynomial spaces and applies them to establish precise eigenvalue error estimates in finite element analysis.

Findings

Sharper error bounds for piecewise polynomial approximation.

Lower bounds for eigenvalue discretization errors in finite element methods.

Analysis of asymptotic convergence behavior of eigenvalues.

Abstract

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.