Analysis of the FEM and DGM for an elliptic problem with a nonlinear Newton boundary condition

Feistauer, Miloslav; Bartoš, Ondřej; Roskovec, Filip; Sändig, Anna-Margarete

  • Proceedings of Equadiff 14, Publisher: Slovak University of Technology in Bratislava, SPEKTRUM STU Publishing(Bratislava), page 127-136

Abstract

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The paper is concerned with the numerical analysis of an elliptic equation in a polygon with a nonlinear Newton boundary condition, discretized by the finite element or discontinuous Galerkin methods. Using the monotone operator theory, it is possible to prove the existence and uniqueness of the exact weak solution and the approximate solution. The main attention is paid to the study of error estimates. To this end, the regularity of the weak solution is investigated and it is shown that due to the boundary corner points, the solution looses regularity in a vicinity of these points. It comes out that the error estimation depends essentially on the opening angle of the corner points and on the parameter defining the nonlinear behaviour of the Newton boundary condition. Theoretical results are compared with numerical experiments confirming a nonstandard behaviour of error estimates.

How to cite

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Feistauer, Miloslav, et al. "Analysis of the FEM and DGM for an elliptic problem with a nonlinear Newton boundary condition." Proceedings of Equadiff 14. Bratislava: Slovak University of Technology in Bratislava, SPEKTRUM STU Publishing, 2017. 127-136. <http://eudml.org/doc/294892>.

@inProceedings{Feistauer2017,
abstract = {The paper is concerned with the numerical analysis of an elliptic equation in a polygon with a nonlinear Newton boundary condition, discretized by the finite element or discontinuous Galerkin methods. Using the monotone operator theory, it is possible to prove the existence and uniqueness of the exact weak solution and the approximate solution. The main attention is paid to the study of error estimates. To this end, the regularity of the weak solution is investigated and it is shown that due to the boundary corner points, the solution looses regularity in a vicinity of these points. It comes out that the error estimation depends essentially on the opening angle of the corner points and on the parameter defining the nonlinear behaviour of the Newton boundary condition. Theoretical results are compared with numerical experiments confirming a nonstandard behaviour of error estimates.},
author = {Feistauer, Miloslav, Bartoš, Ondřej, Roskovec, Filip, Sändig, Anna-Margarete},
booktitle = {Proceedings of Equadiff 14},
keywords = {Elliptic equation, nonlinear Newton boundary condition, monotone operator method, finite element method, discontinuous Galerkin method, regularity and singular behaviour of the solution, error estimation},
location = {Bratislava},
pages = {127-136},
publisher = {Slovak University of Technology in Bratislava, SPEKTRUM STU Publishing},
title = {Analysis of the FEM and DGM for an elliptic problem with a nonlinear Newton boundary condition},
url = {http://eudml.org/doc/294892},
year = {2017},
}

TY - CLSWK
AU - Feistauer, Miloslav
AU - Bartoš, Ondřej
AU - Roskovec, Filip
AU - Sändig, Anna-Margarete
TI - Analysis of the FEM and DGM for an elliptic problem with a nonlinear Newton boundary condition
T2 - Proceedings of Equadiff 14
PY - 2017
CY - Bratislava
PB - Slovak University of Technology in Bratislava, SPEKTRUM STU Publishing
SP - 127
EP - 136
AB - The paper is concerned with the numerical analysis of an elliptic equation in a polygon with a nonlinear Newton boundary condition, discretized by the finite element or discontinuous Galerkin methods. Using the monotone operator theory, it is possible to prove the existence and uniqueness of the exact weak solution and the approximate solution. The main attention is paid to the study of error estimates. To this end, the regularity of the weak solution is investigated and it is shown that due to the boundary corner points, the solution looses regularity in a vicinity of these points. It comes out that the error estimation depends essentially on the opening angle of the corner points and on the parameter defining the nonlinear behaviour of the Newton boundary condition. Theoretical results are compared with numerical experiments confirming a nonstandard behaviour of error estimates.
KW - Elliptic equation, nonlinear Newton boundary condition, monotone operator method, finite element method, discontinuous Galerkin method, regularity and singular behaviour of the solution, error estimation
UR - http://eudml.org/doc/294892
ER -

References

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