Dirk Praetorius
Vienna University of Technology
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Publication
Featured researches published by Dirk Praetorius.
Computers & Mathematics With Applications | 2014
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
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
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
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
Carsten Carstensen; Matthias Maischak; Dirk Praetorius; Ernst P. Stephan
SIAM Journal on Numerical Analysis | 2014
Michael Feischl; Thomas Führer; Dirk Praetorius
{\eta:=\Vert\phi_{h/2}-\phi_h\Vert}
SIAM Journal on Numerical Analysis | 2013
Michael Feischl; Michael Karkulik; Jens Markus Melenk; Dirk Praetorius
Numerische Mathematik | 2010
Samuel Ferraz-Leite; Christoph Ortner; Dirk Praetorius
to estimate the error
SIAM Journal on Scientific Computing | 2007
Carsten Carstensen; Dirk Praetorius
Numerical Algorithms | 2014
Markus Aurada; Michael Ebner; Michael Feischl; Samuel Ferraz-Leite; Thomas Führer; P. Goldenits; Michael Karkulik; Markus Mayr; Dirk Praetorius
{\Vert\phi-\phi_h\Vert}