Several notes on the circumradius condition

Kučera, Václav. "Several notes on the circumradius condition." Applications of Mathematics 61.3 (2016): 287-298. <http://eudml.org/doc/276994>.

@article{Kučera2016,
abstract = {Recently, the so-called circumradius condition (or estimate) was derived, which is a new estimate of the $W^\{1,p\}$-error of linear Lagrange interpolation on triangles in terms of their circumradius. The published proofs of the estimate are rather technical and do not allow clear, simple insight into the results. In this paper, we give a simple direct proof of the $p=\infty $ case. This allows us to make several observations such as on the optimality of the circumradius estimate. Furthermore, we show how the case of general $p$ is in fact nothing more than a simple scaling of the standard $O(h)$ estimate under the maximum angle condition, even for higher order interpolation. This allows a direct interpretation of the circumradius estimate and condition in the context of the well established theory of the maximum angle condition.},
author = {Kučera, Václav},
journal = {Applications of Mathematics},
keywords = {finite element method; a priori error estimate; circumradius condition; Lagrange interpolation; finite element method; a priori error estimate; circumradius condition; Lagrange interpolation},
language = {eng},
number = {3},
pages = {287-298},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Several notes on the circumradius condition},
url = {http://eudml.org/doc/276994},
volume = {61},
year = {2016},
}

TY - JOUR
AU - Kučera, Václav
TI - Several notes on the circumradius condition
JO - Applications of Mathematics
PY - 2016
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 61
IS - 3
SP - 287
EP - 298
AB - Recently, the so-called circumradius condition (or estimate) was derived, which is a new estimate of the $W^{1,p}$-error of linear Lagrange interpolation on triangles in terms of their circumradius. The published proofs of the estimate are rather technical and do not allow clear, simple insight into the results. In this paper, we give a simple direct proof of the $p=\infty $ case. This allows us to make several observations such as on the optimality of the circumradius estimate. Furthermore, we show how the case of general $p$ is in fact nothing more than a simple scaling of the standard $O(h)$ estimate under the maximum angle condition, even for higher order interpolation. This allows a direct interpretation of the circumradius estimate and condition in the context of the well established theory of the maximum angle condition.
LA - eng
KW - finite element method; a priori error estimate; circumradius condition; Lagrange interpolation; finite element method; a priori error estimate; circumradius condition; Lagrange interpolation
UR - http://eudml.org/doc/276994
ER -