Error estimates for a vectorial second-order elliptic eigenproblem by the local l2 projected c0 finite element method

Abstract

In this paper, a theoretical framework with a completely new theory is presented for the general curlcurl-graddiv second-order elliptic eigenproblem when discretized by the recently developed local L2 projected C 0 finite element method. The theoretical framework consists of two Fortin-type interpolations and an Inf-Sup inequality associated with a trilinear curl/div form. From this framework, error estimates are readily derived for the source problem as well as the eigenproblem. The new theory is verified for the local L2 projected C0 finite element method with elementwise linear element L2 projections applied to div and curl operators and the C0 linear elements enriched with some face-bubbles and element-bubbles for the singular solution. The challenging question is whether C0 elements can give spectrally correct approximations when eigenfunctions are singular (not in (H1(Ω ))d space). It is shown that optimal error bounds can be obtained from this theoretical framework with O(hr) for eigenvalues while O(h r) for singular eigenfunctions in (Hr(Ω)) d (d = 2, 3), where r < 1, and that the local L2 projected C0 finite element method is spectrally correct. © 2013 Society for Industrial and Applied Mathematics.