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Dive into the research topics where Dirk Praetorius is active.

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Featured researches published by Dirk Praetorius.


Computers & Mathematics With Applications | 2014

Axioms of adaptivity

Carsten Carstensen; Michael Feischl; Marcus Page; Dirk Praetorius

This paper aims first at a simultaneous axiomatic presentation of the proof of optimal convergence rates for adaptive finite element methods and second at some refinements of particular questions like the avoidance of (discrete) lower bounds, inexact solvers, inhomogeneous boundary data, or the use of equivalent error estimators. Solely four axioms guarantee the optimality in terms of the error estimators. Compared to the state of the art in the temporary literature, the improvements of this article can be summarized as follows: First, a general framework is presented which covers the existing literature on optimality of adaptive schemes. The abstract analysis covers linear as well as nonlinear problems and is independent of the underlying finite element or boundary element method. Second, efficiency of the error estimator is neither needed to prove convergence nor quasi-optimal convergence behavior of the error estimator. In this paper, efficiency exclusively characterizes the approximation classes involved in terms of the best-approximation error and data resolution and so the upper bound on the optimal marking parameters does not depend on the efficiency constant. Third, some general quasi-Galerkin orthogonality is not only sufficient, but also necessary for the R-linear convergence of the error estimator, which is a fundamental ingredient in the current quasi-optimality analysis due to Stevenson 2007. Finally, the general analysis allows for equivalent error estimators and inexact solvers as well as different non-homogeneous and mixed boundary conditions.


Computational Methods in Applied Mathematics Comput | 2011

Efficient implementation of adaptive P1-FEM in Matlab

Stefan Funken; Dirk Praetorius; Philipp Wissgott

Abstract We provide a MATLAB package p1afem for an adaptive P1-finite element method (AFEM). This includes functions for the assembly of the data, different error estimators, and an indicator-based adaptive meshrefining algorithm. Throughout, the focus is on an efficient realization by use of MATLAB built-in functions and vectorization. Numerical experiments underline the efficiency of the code which is observed to be of almost linear complexity with respect to the runtime. Although the scope of this paper is on AFEM, the general ideas can be understood as a guideline for writing efficient MATLAB code.


SIAM Journal on Scientific Computing | 2005

Averaging Techniques for the Effective Numerical Solution of Symm's Integral Equation of the First Kind

Carsten Carstensen; Dirk Praetorius

Averaging techniques for finite element error control, occasionally called ZZ estimators for the gradient recovery, enjoy a high popularity in engineering because of their striking simplicity and universality: One does not even require a PDE to apply the nonexpensive post-processing routines. Recently, averaging techniques have been mathematically proved to be reliable and efficient for various applications of the finite element method. This paper establishes a class of averaging error estimators for boundary integral methods. Symms integral equation of the first kind with a nonlocal single-layer integral operator serves as a model equation studied both theoretically and numerically. We introduce four new error estimators which are proven to be reliable and efficient up to terms of higher order. The higher-order terms depend on the regularity of the exact solution. Several numerical experiments illustrate the theoretical results and show that the [normally unknown] error is sharply estimated by the proposed estimators, i.e., error and estimators almost coincide.


Computing | 2008

Simple a posteriori error estimators for the h -version of the boundary element method

Samuel Ferraz-Leite; Dirk Praetorius

The h-h/2-strategy is one well-known technique for the a posteriori error estimation for Galerkin discretizations of energy minimization problems. One considers


Numerische Mathematik | 2004

Residual-based a posteriori error estimate for hypersingular equation on surfaces

Carsten Carstensen; Matthias Maischak; Dirk Praetorius; Ernst P. Stephan


SIAM Journal on Numerical Analysis | 2014

Adaptive FEM with Optimal Convergence Rates for a Certain Class of Nonsymmetric and Possibly Nonlinear Problems

Michael Feischl; Thomas Führer; Dirk Praetorius

{\eta:=\Vert\phi_{h/2}-\phi_h\Vert}


SIAM Journal on Numerical Analysis | 2013

Quasi-optimal Convergence Rate for an Adaptive Boundary Element Method

Michael Feischl; Michael Karkulik; Jens Markus Melenk; Dirk Praetorius


Numerische Mathematik | 2010

Convergence of simple adaptive Galerkin schemes based on h − h /2 error estimators

Samuel Ferraz-Leite; Christoph Ortner; Dirk Praetorius

to estimate the error


SIAM Journal on Scientific Computing | 2007

Averaging Techniques for the A Posteriori BEM Error Control for a Hypersingular Integral Equation in Two Dimensions

Carsten Carstensen; Dirk Praetorius


Numerical Algorithms | 2014

HILBERT -- a MATLAB implementation of adaptive 2D-BEM

Markus Aurada; Michael Ebner; Michael Feischl; Samuel Ferraz-Leite; Thomas Führer; P. Goldenits; Michael Karkulik; Markus Mayr; Dirk Praetorius

{\Vert\phi-\phi_h\Vert}

Collaboration


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Michael Feischl

Vienna University of Technology

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Michael Karkulik

Pontifical Catholic University of Chile

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Thomas Führer

Pontifical Catholic University of Chile

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Markus Aurada

Vienna University of Technology

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Jens Markus Melenk

Vienna University of Technology

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Carsten Carstensen

Humboldt University of Berlin

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Marcus Page

Vienna University of Technology

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Samuel Ferraz-Leite

Vienna University of Technology

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Florian Bruckner

Vienna University of Technology

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