Super-Closeness Between the Ritz Projection and the Finite Element Solution for Some Elliptic Problems
Year: 2020
Author: Ruosi Liang, Yue Yan, Wenbin Chen, Yanqiu Wang
Communications in Computational Physics, Vol. 28 (2020), Iss. 2 : pp. 803–826
Abstract
We prove the super-closeness between the finite element solution and the Ritz projection for some second order and fourth order elliptic equations in both the $H^1$ and the $L^2$ norms. For the fourth order problem, a Ciarlet-Raviart type mixed formulation is used in the analysis. The main tool in the proof is a negative norm estimate of the Ritz projection, which requires $H^{q+1}$ regularity for second order elliptic equations. Therefore, the analysis is done on a domain Ω with smooth boundary, and hence we only consider the pure Neumann boundary problems which can be discretized naturally on such domains, if ignoring the effect of numerical integrals. For the fourth order problem, our results amend the gap between the theoretical estimates and the numerical examples in a previous work [22].
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/cicp.OA-2019-0154
Communications in Computational Physics, Vol. 28 (2020), Iss. 2 : pp. 803–826
Published online: 2020-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 24
Keywords: Finite element Ritz projection negative norm estimate super-closeness estimate.