Dirk Praetorius

Bio: Dirk Praetorius is an academic researcher from Vienna University of Technology. The author has contributed to research in topics: Finite element method & Estimator. The author has an hindex of 29, co-authored 199 publications receiving 3188 citations.

Axioms of adaptivity

Abstract: 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.

Efficient implementation of adaptive P1-FEM in Matlab

Abstract: We provide a Matlab implementation of an adaptive P1-finite element method (AFEM). This includes functions for the assembly of the data, different error estimators, and an indicator-based adaptive mesh-refining 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.

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

Abstract: We analyze adaptive mesh-refining algorithms for conforming finite element discretizations of certain nonlinear second-order partial differential equations. We allow continuous polynomials of arbitrary but fixed polynomial order. The adaptivity is driven by the residual error estimator. We prove convergence even with optimal algebraic convergence rates. In particular, our analysis covers general linear second-order elliptic operators. Unlike prior works for linear nonsymmetric operators, our analysis avoids the interior node property for the refinement, and the differential operator has to satisfy a G\rarding inequality only. If the differential operator is uniformly elliptic, no additional assumption on the initial mesh is posed.

On 2D Newest Vertex Bisection: Optimality of Mesh-Closure and H 1-Stability of L 2-Projection

Abstract: Newest vertex bisection (NVB) is a popular local mesh-refinement strategy for regular triangulations that consist of simplices. For the 2D case, we prove that the mesh-closure step of NVB, which preserves regularity of the triangulation, is quasi-optimal and that the corresponding L 2-projection onto lowest-order Courant finite elements (P1-FEM) is always H 1-stable. Throughout, no additional assumptions on the initial triangulation are imposed. Our analysis thus improves results of Binev et al. (Numer. Math. 97(2):219–268, 2004), Carstensen (Constr. Approx. 20(4):549–564, 2004), and Stevenson (Math. Comput. 77(261):227–241, 2008) in the sense that all assumptions of their theorems are removed. Consequently, our results relax the requirements under which adaptive finite element schemes can be mathematically guaranteed to convergence with quasi-optimal rates.

Solving Ordinary Differential Equations

Abstract: Initial-value ordinary differential equation solution via variable order Adams method (SIVA/DIVA) package is collection of subroutines for solution of nonstiff ordinary differential equations. There are versions for single-precision and double-precision arithmetic. Requires fewer evaluations of derivatives than other variable-order Adams predictor/ corrector methods. Option for direct integration of second-order equations makes integration of trajectory problems significantly more efficient. Written in FORTRAN 77.

On the Navier-Stokes equations

Abstract: A criterion is given for the convergence of numerical solutions of the Navier-Stokes equations in two dimensions under steady conditions. The criterion applies to all cases, of steady viscous flow in two dimensions and shows that if the local ' mesh Reynolds number ', based on the size of the mesh used in the solution, exceeds a certain fixed value, the numerical solution will not converge.

Heat and Mass Transfer

Abstract: Thermophysical Properties Research Literature Retrieval Guide Edited by Y. S. Touloukian, J. K. Gerritsen and N. Y. Moore Second edition, revised and expanded. Book 1: Pp. xxi + 819. Book 2: Pp.621. Book 3: Pp. ix + 1315. (New York: Plenum Press, 1967.) n.p.