Sharp Error Bounds for Piecewise Polynomial Approximation: Revisit and Application to Elliptic PDE Eigenvalue Computation
Year: 2015
Type: preprint
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.
Source: arXiv (Cornell University)
Authors: Xie, Hehu, Zhang, Zhimin
Language: English
Subfield: Computational Mechanics
Field: Engineering
Domain: Physical Sciences
Open Access status: green
Awards: G2660812258, G2746842099, G3715460442, G3946751353, 若干具有弱导数的新型计算方法的梯度重构, G5850044563, G5934128047, 工业结晶过程的数值模拟, 相场方程的高效数值算法, Spectral and spectral collocation methods for Hamiltonian systems, G8598914340, Superconvergent post-processing of some newly developed numerical methods with weak derivatives, G8964106685
OpenAlex ID: W157291749