$H^1$-norm error estimate for a nonstandard finite element approximation of second-order linear elliptic PDEs in non-divergence form
Numerical Analysis
2019-10-01 v1 Numerical Analysis
Abstract
This paper establishes the optimal -norm error estimate for a nonstandard finite element method for approximating strong solutions of second order linear elliptic PDEs in non-divergence form with continuous coefficients. To circumvent the difficulty of lacking an effective duality argument for this class of PDEs, a new analysis technique is introduced; the crux of it is to establish an -norm stability estimate for the finite element approximation operator which mimics a similar estimate for the underlying PDE operator recently established by the authors and its proof is based on a freezing coefficient technique and a topological argument. Moreover, both the -norm stability and error estimate also hold for the linear finite element method.
Keywords
Cite
@article{arxiv.1909.13803,
title = {$H^1$-norm error estimate for a nonstandard finite element approximation of second-order linear elliptic PDEs in non-divergence form},
author = {Xiaobing Feng and Stefan Schnake},
journal= {arXiv preprint arXiv:1909.13803},
year = {2019}
}