ERROR ESTIMATES FOR A VECTORIAL SECOND-ORDER ELLIPTIC EIGENPROBLEM BY THE LOCAL L-2 PROJECTED C-0 FINITE ELEMENT METHOD

作者：Duan Huo Yuan^*; Lin Ping; Tan Roger C E

来源：SIAM Journal on Numerical Analysis, 2013, 51(3): 1678-1714.

DOI：10.1137/100812136

• curlcurl-graddiv eigenproblem
• singular solution
• L-2 projected C-0 finite element method
• Fortin-type interpolation
• Inf-Sup inequality
• spectral correct approximation
• error estimates

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 L-2 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 L-2 projected C-0 finite element method with elementwise linear element L-2 projections applied to div and curl operators and the C-0 linear elements enriched with some face-bubbles and element-bubbles for the singular solution. The challenging question is whether C-0 elements can give spectrally correct approximations when eigenfunctions are singular (not in (H-1(Omega))(d) space). It is shown that optimal error bounds can be obtained from this theoretical framework with O(h(2r)) for eigenvalues while O(h(r)) for singular eigenfunctions in (H-r(Omega))(d) (d = 2, 3), where r %26lt; 1, and that the local L-2 projected C-0 finite element method is spectrally correct.

• 出版日期2013
• 单位南开大学