On the effect of numerical integration in the finite element solution of an elliptic problem with a nonlinear Newton boundary condition

Ondřej Bartoš; Miloslav Feistauer; Filip Roskovec

Applications of Mathematics (2019)

  • Volume: 64, Issue: 2, page 129-167
  • ISSN: 0862-7940

Abstract

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This paper is concerned with the analysis of the finite element method for the numerical solution of an elliptic boundary value problem with a nonlinear Newton boundary condition in a two-dimensional polygonal domain. The weak solution loses regularity in a neighbourhood of boundary singularities, which may be at corners or at roots of the weak solution on edges. The main attention is paid to the study of error estimates. It turns out that the order of convergence is not dampened by the nonlinearity if the weak solution is nonzero on a large part of the boundary. If the weak solution is zero on the whole boundary, the nonlinearity only slows down the convergence of the function values but not the convergence of the gradient. The same analysis is carried out for approximate solutions obtained by numerical integration. The theoretical results are verified by numerical experiments.

How to cite

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Bartoš, Ondřej, Feistauer, Miloslav, and Roskovec, Filip. "On the effect of numerical integration in the finite element solution of an elliptic problem with a nonlinear Newton boundary condition." Applications of Mathematics 64.2 (2019): 129-167. <http://eudml.org/doc/294831>.

@article{Bartoš2019,
abstract = {This paper is concerned with the analysis of the finite element method for the numerical solution of an elliptic boundary value problem with a nonlinear Newton boundary condition in a two-dimensional polygonal domain. The weak solution loses regularity in a neighbourhood of boundary singularities, which may be at corners or at roots of the weak solution on edges. The main attention is paid to the study of error estimates. It turns out that the order of convergence is not dampened by the nonlinearity if the weak solution is nonzero on a large part of the boundary. If the weak solution is zero on the whole boundary, the nonlinearity only slows down the convergence of the function values but not the convergence of the gradient. The same analysis is carried out for approximate solutions obtained by numerical integration. The theoretical results are verified by numerical experiments.},
author = {Bartoš, Ondřej, Feistauer, Miloslav, Roskovec, Filip},
journal = {Applications of Mathematics},
keywords = {elliptic equation; nonlinear Newton boundary condition; weak solution; finite element discretization; numerical integration; error estimation; effect of numerical integration},
language = {eng},
number = {2},
pages = {129-167},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the effect of numerical integration in the finite element solution of an elliptic problem with a nonlinear Newton boundary condition},
url = {http://eudml.org/doc/294831},
volume = {64},
year = {2019},
}

TY - JOUR
AU - Bartoš, Ondřej
AU - Feistauer, Miloslav
AU - Roskovec, Filip
TI - On the effect of numerical integration in the finite element solution of an elliptic problem with a nonlinear Newton boundary condition
JO - Applications of Mathematics
PY - 2019
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 64
IS - 2
SP - 129
EP - 167
AB - This paper is concerned with the analysis of the finite element method for the numerical solution of an elliptic boundary value problem with a nonlinear Newton boundary condition in a two-dimensional polygonal domain. The weak solution loses regularity in a neighbourhood of boundary singularities, which may be at corners or at roots of the weak solution on edges. The main attention is paid to the study of error estimates. It turns out that the order of convergence is not dampened by the nonlinearity if the weak solution is nonzero on a large part of the boundary. If the weak solution is zero on the whole boundary, the nonlinearity only slows down the convergence of the function values but not the convergence of the gradient. The same analysis is carried out for approximate solutions obtained by numerical integration. The theoretical results are verified by numerical experiments.
LA - eng
KW - elliptic equation; nonlinear Newton boundary condition; weak solution; finite element discretization; numerical integration; error estimation; effect of numerical integration
UR - http://eudml.org/doc/294831
ER -

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