Error estimates for DGFE solutions are well investigated if one assumes
that the exact solution is sufficiently regular. In this article,
we consider a Dirichlet and a mixed boundary value problem for a
linear elliptic equation in a polygon. It is well known that the
first derivatives of the solutions develop singularities near reentrant
corner points or points where the boundary conditions change. On
the basis of the regularity results formulated in Sobolev--Slobodetskii
spaces and weighted spaces of Kondratiev type, we prove error estimates
of higher order for DGFE solutions using a suitable graded mesh refinement
near boundary singular points. The main tools are as follows: regularity
investigation for the exact solution relying on general results for
elliptic boundary value problems, error analysis for the interpolation
in Sobolev--Slobodetskii spaces, and error estimates for DGFE solutions
on special graded refined meshes combined with estimates in weighted
Sobolev spaces. Our main result is that there exist a local grading
of the mesh and a piecewise interpolation by polynoms of higher degree
such that we will get the same order O (ha) of approximation as in
the smooth case. � 2011 Wiley Periodicals, Inc. Numer Mehods Partial
Differential Eq, 2012
%0 Journal Article
%1 feistauer2012graded
%A Feistauer, Miloslav
%A Sändig, Anna-Margarete
%D 2012
%I Wiley Subscription Services, Inc., A Wiley Company
%J Numerical Methods for Partial Differential Equations
%K Galerkin Sobolev Sobolev--Slobodetskii boundary discontinuous elliptic graded mesh method, problems, refinement, spaces spaces, value vorlaeufig weighted
%N 4
%P 1124--1151
%R 10.1002/num.20668
%T Graded mesh refinement and error estimates of higher order for DGFE
solutions of elliptic boundary value problems in polygons
%U http://dx.doi.org/10.1002/num.20668
%V 28
%X Error estimates for DGFE solutions are well investigated if one assumes
that the exact solution is sufficiently regular. In this article,
we consider a Dirichlet and a mixed boundary value problem for a
linear elliptic equation in a polygon. It is well known that the
first derivatives of the solutions develop singularities near reentrant
corner points or points where the boundary conditions change. On
the basis of the regularity results formulated in Sobolev--Slobodetskii
spaces and weighted spaces of Kondratiev type, we prove error estimates
of higher order for DGFE solutions using a suitable graded mesh refinement
near boundary singular points. The main tools are as follows: regularity
investigation for the exact solution relying on general results for
elliptic boundary value problems, error analysis for the interpolation
in Sobolev--Slobodetskii spaces, and error estimates for DGFE solutions
on special graded refined meshes combined with estimates in weighted
Sobolev spaces. Our main result is that there exist a local grading
of the mesh and a piecewise interpolation by polynoms of higher degree
such that we will get the same order O (ha) of approximation as in
the smooth case. � 2011 Wiley Periodicals, Inc. Numer Mehods Partial
Differential Eq, 2012
@article{feistauer2012graded,
abstract = {Error estimates for DGFE solutions are well investigated if one assumes
that the exact solution is sufficiently regular. In this article,
we consider a Dirichlet and a mixed boundary value problem for a
linear elliptic equation in a polygon. It is well known that the
first derivatives of the solutions develop singularities near reentrant
corner points or points where the boundary conditions change. On
the basis of the regularity results formulated in Sobolev--Slobodetskii
spaces and weighted spaces of Kondratiev type, we prove error estimates
of higher order for DGFE solutions using a suitable graded mesh refinement
near boundary singular points. The main tools are as follows: regularity
investigation for the exact solution relying on general results for
elliptic boundary value problems, error analysis for the interpolation
in Sobolev--Slobodetskii spaces, and error estimates for DGFE solutions
on special graded refined meshes combined with estimates in weighted
Sobolev spaces. Our main result is that there exist a local grading
of the mesh and a piecewise interpolation by polynoms of higher degree
such that we will get the same order O (ha) of approximation as in
the smooth case. � 2011 Wiley Periodicals, Inc. Numer Mehods Partial
Differential Eq, 2012},
added-at = {2018-07-20T10:54:15.000+0200},
author = {Feistauer, Miloslav and S{\"a}ndig, Anna-Margarete},
biburl = {https://puma.ub.uni-stuttgart.de/bibtex/29db316fbe639db8fe7dbd32263b10ad6/mhartmann},
doi = {10.1002/num.20668},
interhash = {3b3eb9b7f23cb0657ebafecd547be4d8},
intrahash = {9db316fbe639db8fe7dbd32263b10ad6},
issn = {1098-2426},
journal = {Numerical Methods for Partial Differential Equations},
keywords = {Galerkin Sobolev Sobolev--Slobodetskii boundary discontinuous elliptic graded mesh method, problems, refinement, spaces spaces, value vorlaeufig weighted},
number = 4,
pages = {1124--1151},
publisher = {Wiley Subscription Services, Inc., A Wiley Company},
timestamp = {2018-07-20T08:54:15.000+0200},
title = {Graded mesh refinement and error estimates of higher order for DGFE
solutions of elliptic boundary value problems in polygons},
url = {http://dx.doi.org/10.1002/num.20668},
volume = 28,
year = 2012
}