PUMA publications for /user/mhartmann/method,https://puma.ub.uni-stuttgart.de/user/mhartmann/method,PUMA RSS feed for /user/mhartmann/method,2019-06-17T11:07:38+02:00Graded mesh refinement and error estimates of higher order for DGFE
solutions of elliptic boundary value problems in polygonshttps://puma.ub.uni-stuttgart.de/bibtex/29db316fbe639db8fe7dbd32263b10ad6/mhartmannmhartmann2018-07-20T10:54:15+02:00Galerkin Sobolev Sobolev--Slobodetskii boundary discontinuous elliptic graded mesh method, problems, refinement, spaces spaces, value vorlaeufig weighted <span class="authorEditorList"><span itemtype="http://schema.org/Person" itemscope="itemscope" itemprop="author"><a title="Miloslav Feistauer" itemprop="url" href="/person/13b3eb9b7f23cb0657ebafecd547be4d8/author/0"><span itemprop="name">M. Feistauer</span></a></span>, und <span itemtype="http://schema.org/Person" itemscope="itemscope" itemprop="author"><a title="Anna-Margarete Sändig" itemprop="url" href="/person/13b3eb9b7f23cb0657ebafecd547be4d8/author/1"><span itemprop="name">A. Sändig</span></a></span>. </span><span itemtype="http://schema.org/PublicationIssue" itemscope="itemscope" itemprop="isPartOf"><span itemtype="http://schema.org/Periodical" itemscope="itemscope" itemprop="isPartOf"><span itemprop="name"><em>Numerical Methods for Partial Differential Equations</em></span></span> <em><span itemtype="http://schema.org/PublicationVolume" itemscope="itemscope" itemprop="isPartOf"><span itemprop="volumeNumber">28 </span></span>(<span itemprop="issueNumber">4</span>):
<span itemprop="pagination">1124--1151</span></em> </span>(<em><span>2012<meta content="2012" itemprop="datePublished"/></span></em>)Fri Jul 20 10:54:15 CEST 2018Numerical Methods for Partial Differential Equations41124--1151Graded mesh refinement and error estimates of higher order for DGFE
solutions of elliptic boundary value problems in polygons282012Galerkin Sobolev Sobolev--Slobodetskii boundary discontinuous elliptic graded mesh method, problems, refinement, spaces spaces, value vorlaeufig weighted 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, 2012An adaptive wavelet space-time SUPG method for hyperbolic conservation
lawshttps://puma.ub.uni-stuttgart.de/bibtex/24b6636216b66dfdfd50ca53bb93685de/mhartmannmhartmann2018-07-20T10:54:15+02:00(SUPG), Galerkin, Petrove-Galerkin adaptive conservation continuous discontinuous hyperbolic laws, method, postprocessing spectral streamline upwind viscosity, vorlaeufig wavelet <span class="authorEditorList"><span itemtype="http://schema.org/Person" itemscope="itemscope" itemprop="author"><a title="Hadi Minbashian" itemprop="url" href="/person/1c8972e3ab25e760174ed7741330758be/author/0"><span itemprop="name">H. Minbashian</span></a></span>, <span itemtype="http://schema.org/Person" itemscope="itemscope" itemprop="author"><a title="Hojatolah Adibi" itemprop="url" href="/person/1c8972e3ab25e760174ed7741330758be/author/1"><span itemprop="name">H. Adibi</span></a></span>, und <span itemtype="http://schema.org/Person" itemscope="itemscope" itemprop="author"><a title="Mehdi Dehghan" itemprop="url" href="/person/1c8972e3ab25e760174ed7741330758be/author/2"><span itemprop="name">M. Dehghan</span></a></span>. </span><span itemtype="http://schema.org/PublicationIssue" itemscope="itemscope" itemprop="isPartOf"><span itemtype="http://schema.org/Periodical" itemscope="itemscope" itemprop="isPartOf"><span itemprop="name"><em>Numerical Methods for Partial Differential Equations</em></span></span> <em><span itemtype="http://schema.org/PublicationVolume" itemscope="itemscope" itemprop="isPartOf"><span itemprop="volumeNumber">33 </span></span>(<span itemprop="issueNumber">6</span>):
<span itemprop="pagination">2062-2089</span></em> </span>(<em><span>2017<meta content="2017" itemprop="datePublished"/></span></em>)Fri Jul 20 10:54:15 CEST 2018Numerical Methods for Partial Differential Equations62062-2089An adaptive wavelet space-time SUPG method for hyperbolic conservation
laws332017(SUPG), Galerkin, Petrove-Galerkin adaptive conservation continuous discontinuous hyperbolic laws, method, postprocessing spectral streamline upwind viscosity, vorlaeufig wavelet This article concerns with incorporating wavelet bases into existing
streamline upwind Petrov-Galerkin (SUPG) methods for the numerical
solution of nonlinear hyperbolic conservation laws which are known
to develop shock solutions. Here, we utilize an SUPG formulation
using continuous Galerkin in space and discontinuous Galerkin in
time. The main motivation for such a combination is that these methods
have good stability properties thanks to adding diffusion in the
direction of streamlines. But they are more expensive than explicit
semidiscrete methods as they have to use space-time formulations.
Using wavelet bases we maintain the stability properties of SUPG
methods while we reduce the cost of these methods significantly through
natural adaptivity of wavelet expansions. In addition, wavelet bases
have a hierarchical structure. We use this property to numerically
investigate the hierarchical addition of an artificial diffusion
for further stabilization in spirit of spectral diffusion. Furthermore,
we add the hierarchical diffusion only in the vicinity of discontinuities
using the feature of wavelet bases in detection of location of discontinuities.
Also, we again use the last feature of the wavelet bases to perform
a postprocessing using a denosing technique based on a minimization
formulation to reduce Gibbs oscillations near discontinuities while
keeping other regions intact. Finally, we show the performance of
the proposed combination through some numerical examples including
Burgers�, transport, and wave equations as well as systems of shallow
water equations.� 2017 Wiley Periodicals, Inc. Numer Methods Partial
Differential Eq 33: 2062�2089, 2017