Veröffentlichungen

@misc {KoRoSi:13a,
author = {Kristina Kohls and Arnd Rösch and Kunibert G. Siebert} ,
title = {A Posteriori Error Analysis of Optimal Control Problems with Control Constraints} ,
year = {2013} }

@article {dunecosmo:12,
author = {Brdar, S. and Baldauf, M. and Dedner, A. and Klöfkorn, R.} ,
title = {Comparison of dynamical cores for NWP models: comparison of COSMO and Dune} ,
journal = {Theoretical and Computational Fluid Dynamics} ,
publisher = {Springer-Verlag} ,
year = {2012} ,
pages = {1-20} ,
url = {http://dx.doi.org/10.1007/s00162-012-0264-z} ,
doi = {10.1007/s00162-012-0264-z} }

@article {cdg2:12,
author = {Brdar, S. and Dedner, A. and Klöfkorn, R.} ,
title = {Compact and stable Discontinuous Galerkin methods for convection-diffusion problems.} ,
journal = {SIAM J. Sci. Comput.} ,
year = {2012} ,
volume = {34} ,
number = {1} ,
url = {http://dx.doi.org/10.1137/100817528} ,
doi = {10.1137/100817528} }

@book {dum:11,
author = {Dedner, A. and Flemisch, B. and Klöfkorn, R.} ,
title = {Advances in DUNE} ,
publisher = {Springer} ,
year = {2012} }

@incollection {dumproc:11,
author = {Dedner, A. and Klöfkorn, R. and Nolte, M. and Ohlberger, M.} ,
editor = {Dedner, A. and Flemisch, B. and Klöfkorn, R.} ,
title = {Dune-Fem: A General Purpose Discretization Toolbox for Parallel and Adaptive Scientific Computing} ,
booktitle = {Advances in DUNE} ,
publisher = {Springer Berlin Heidelberg} ,
year = {2012} ,
pages = {17-31} ,
url = {http://dx.doi.org/10.1007/978-3-642-28589-9_2} ,
doi = {10.1007/978-3-642-28589-9_2} }

@inproceedings {dgimpl:12,
author = {Klöfkorn, R.} ,
editor = {A. Handlovicova et al.} ,
title = {Efficient Matrix-Free Implementation of Discontinuous Galerkin Methods for Compressible Flow Problems} ,
booktitle = {Proceedings of the ALGORITMY 2012} ,
year = {2012} ,
pages = {11-21} }

@incollection {perf:11,
author = {Klöfkorn, R. and Nolte, M.} ,
editor = {Dedner, A. and Flemisch, B. and Klöfkorn, R.} ,
title = {Performance Pitfalls in the Dune Grid Interface} ,
booktitle = {Advances in DUNE} ,
publisher = {Springer Berlin Heidelberg} ,
year = {2012} ,
pages = {45-58} ,
url = {http://dx.doi.org/10.1007/978-3-642-28589-9_4} ,
doi = {10.1007/978-3-642-28589-9_4} }

@incollection {KoRoSi:12,
author = {Kristina Kohls and Arnd Rösch and Kunibert G. Siebert} ,
editor = {Leugering et al.} ,
title = {A Posteriori Error Estimators for Control Constrained Optimal Control Problems} ,
booktitle = {Constrained Optimiziation and Optimal Control for Partial Differential Equations} ,
publisher = {Springer} ,
year = {2012} ,
volume = {160} ,
pages = {431-443} ,
url = {http://dx.doi.org/10.1007/978-3-0348-0133-1_22} ,
doi = {10.1007/978-3-0348-0133-1_22} }

Abstract: In this note we present a framework for the a posteriori error analysis of control constrained optimal control problems with linear PDE constraints. It is solely based on reliable and efficient error estimators for the corresponding linear state and adjoint equations. We show that the sum of these estimators gives a reliable and efficient estimator for the optimal control problem.

@electronic {KrMoSchSi:12,
author = {Christian Kreuzer and Christian Möller and Alfred Schmidt and Kunibert G. Siebert} ,
title = {Design and Convergence Analysis for an Adaptive Discretization of the Heat Equation} ,
journal = {IMA Journal of Numerical Analysis} ,
year = {2012} ,
note = {Online First} ,
url = {http://dx.doi.org/10.1093/imanum/drr026} ,
doi = {10.1093/imanum/drr026} }

Abstract: We derive an algorithm for the adaptive approximation of solutions to parabolic equations. It is based on adaptive finite elements in space and the implicit Euler discretization in time with adaptive time-step sizes. We prove that, given a positive tolerance for the error, the adaptive algorithm reaches the final time with a space–time error between continuous and discrete solution that is below the given tolerance. Numerical experiments reveal a more than competitive performance of our algorithm ASTFEM (adaptive space–time finite element method).

@incollection {Siebert:12,
author = {Kunibert G. Siebert} ,
title = {Mathematically Founded Design of Adaptive Finite Element Software} ,
booktitle = {Multiscale and Adaptivity: Modelling, Numerics and Applications} ,
publisher = {Springer} ,
year = {2012} ,
volume = {2040} ,
pages = {227-309} ,
url = {http://dx.doi.org/10.1007/978-3-642-24079-9_4} ,
doi = {10.1007/978-3-642-24079-9_4} }

Abstract: In these lecture notes we derive from the mathematical concepts of adaptive finite element methods basic design principles of adaptive finite element software. We introduce finite element spaces, discuss local refinement of simplical grids, the assemblage and structure of the discrete linear system, the computation of the error estimator, and common adaptive strategies. The mathematical discussion naturally leads to appropriate data structures and efficient algorithms for the implementation. The theoretical part is complemented by exercises giving an introduction to the implementation of solvers for linear and nonlinear problems in the adaptive finite element toolbox ALBERTA.

@article {GaspozMorin:2009,
author = {Fernando D. Gaspoz and Pedro Morin} ,
title = {Convergence rates for adaptive finite elements} ,
journal = {IMA J. Numer. Anal.} ,
publisher = {Oxford University Press} ,
year = {2009} ,
volume = {29} ,
number = {4} ,
pages = {917--936} }

@phdthesis {heinecj2003,
author = {Heine, C.-J.} ,
title = {Computations of form and stability of rotating drops with finite elements} ,
school = {RWTH Aachen} ,
year = {2003} }

@incollection {simlab:10,
author = {Brdar, S. and Dedner, A. and Klöfkorn, R.} ,
editor = {I. Kondov et al.} ,
title = {Compact and Stable Discontinuous Galerkin Methods with Application to Atmospheric Flows} ,
booktitle = {Computational Methods in Science and Engineering: Proceedings of the Workshop SimLabs@KIT} ,
publisher = {KIT Scientific Publishing} ,
year = {2011} ,
pages = {109-116} ,
url = {http://dx.doi.org/10.5445/KSP/1000023323} ,
doi = {10.5445/KSP/1000023323} }

@incollection {femgeo:11,
author = {Brdar, S. and Dedner, A. and Klöfkorn, R. and Kränkel, M. and Kröner, D.} ,
editor = {E. Krause et al.} ,
title = {Simulation of Geophysical Problems with DUNE-FEM} ,
booktitle = {Computational Science and High Performance Computing IV} ,
publisher = {Springer} ,
year = {2011} ,
volume = {115} ,
pages = {93-106} ,
url = {http://dx.doi.org/10.1007/978-3-642-17770-5_8} ,
doi = {10.1007/978-3-642-17770-5_8} }

@inproceedings {modred:11,
author = { Dedner, A. and Fein, M. and Klöfkorn, R. and Kröner, D. and Lebiedz, D. and Siehr, J. and Unger, J.} ,
title = {On the computation of slow manifolds in chemical kinetics via optimization and their use as reduced models in reactive flow systems.} ,
year = {2011} ,
note = {to appear} }

@article {limiter:11,
author = {Dedner, A. and Klöfkorn, R.} ,
title = {A Generic Stabilization Approach for Higher Order Discontinuous Galerkin Methods for Convection Dominated Problems} ,
journal = {J. Sci. Comput.} ,
year = {2011} ,
volume = {47} ,
number = {3} ,
pages = {365-388} ,
url = {http://dx.doi.org/10.1007/s10915-010-9448-0} ,
doi = {10.1007/s10915-010-9448-0} }

@incollection {fvca6_bench:11,
author = {Eymard, R. and Henry, G. and Herbin, R. and Hubert, F. and Klöfkorn, R. and Manzini, G.} ,
editor = {Fort, J. and Fürst, J. and Halama, J. and Herbin, R. and Hubert, F.} ,
title = {3D Benchmark on Discretization Schemes for Anisotropic Diffusion Problems on General Grids} ,
booktitle = {Finite Volumes for Complex Applications VI Problems & Perspectives} ,
publisher = {Springer Berlin Heidelberg} ,
year = {2011} ,
volume = {4} ,
pages = {895-930} ,
url = {http://dx.doi.org/10.1007/978-3-642-20671-9_89} ,
doi = {10.1007/978-3-642-20671-9_89} }

@incollection {fvca6_cdg2:11,
author = {Klöfkorn, R.} ,
editor = {Fort, J. and Fürst, J. and Halama, J. and Herbin, R. and Hubert, F.} ,
title = {Benchmark 3D: The Compact Discontinuous Galerkin 2 Scheme} ,
booktitle = {Finite Volumes for Complex Applications VI Problems & Perspectives} ,
publisher = {Springer Berlin Heidelberg} ,
year = {2011} ,
volume = {4} ,
pages = {1023-1033} ,
url = {http://dx.doi.org/10.1007/978-3-642-20671-9_100} ,
doi = {10.1007/978-3-642-20671-9_100} }

@article {KreuzerSiebert:11,
author = {Christian Kreuzer and Kunibert G. Siebert} ,
title = {Decay Rates of Adaptive Finite Elements with Dörfler Marking} ,
journal = {Numerische Mathematik} ,
year = {2011} ,
volume = {117} ,
number = {4} ,
pages = {679-716} ,
url = {http://dx.doi.org/10.1007/s00211-010-0324-5} ,
doi = {10.1007/s00211-010-0324-5} }

Abstract: We investigate the decay rate for an adaptive finite element discretization of a second order linear, symmetric, elliptic PDE. We allow for any kind of estimator that is locally equivalent to the standard residual estimator. This includes in particular hierarchical estimators, estimators based on the solution of local problems, estimators based on local averaging, equilibrated residual estimators, the ZZ-estimator, etc. The adaptive method selects elements for refinement with Dörfler marking and performs a minimal refinement in that no interior node property is needed. Based on the local equivalence to the residual estimator we prove an error reduction property. In combination with minimal Dörfler marking this yields an optimal decay rate in terms of degrees of freedom.

@article {Siebert:11,
author = {Kunibert G. Siebert} ,
title = {A Convergence Proof for Adaptive Finite Elements without Lower Bound} ,
journal = {IMA Journal of Numerical Analysis} ,
year = {2011} ,
volume = {31} ,
number = {3} ,
pages = {947-970} ,
url = {http://imajna.oxfordjournals.org/content/31/3/947.abstract} }

Abstract: We analyse the adaptive finite-element approximation to solutions of partial differential equations in variational formulation. Assuming well-posedness of the continuous problem and requiring only basic properties of the adaptive algorithm, we prove convergence of the sequence of discrete solutions to the true one. The proof is based on the ideas by Morin, Siebert and Veeser but replaces local efficiency of the estimator by a local density property of the adaptively generated finite-element spaces. As a result, estimators without a discrete lower bound are also included in our theory. The assumptions of the presented framework are fulfilled by a large class of important applications, estimators and adaptive strategies.

@incollection {cdg:10,
author = {Brdar, S. and Dedner, A. and Klöfkorn, R.} ,
title = {CDG Method for Navier-Stokes Equations} ,
booktitle = {Proc. of the 13th International Conference on Hyperbolic Problems: Theory, Numerics, Applications} ,
year = {2010} ,
note = {to appear} }

@article {heinecjDeckelnickDziukElliottHeine2010,
author = {Deckelnick, K. and Dziuk, G. and Elliott, C. M. and Heine, C.-J.} ,
title = {An $h$-narrow band finite-element method for elliptic equations on implicit surfaces} ,
journal = { IMA J. Numer. Anal.} ,
year = {2010} ,
volume = {30} ,
number = {2} ,
pages = {351-376} }

@incollection {hlrs:09,
author = {Dedner, A. and Klöfkorn, R. and Kröner, D.} ,
editor = {W. Nagel et al.} ,
title = {Higher Order Adaptive and Parallel Simulations Including Dynamic Load Balancing with the Software Package DUNE} ,
booktitle = {High Performance Computing in Science and Engineering '09} ,
publisher = {Springer} ,
year = {2010} ,
pages = {229-239} ,
url = {http://dx.doi.org/10.1007/978-3-642-04665-0_16} ,
doi = {10.1007/978-3-642-04665-0_16} }

@article {dune:Fem,
author = {Dedner, A. and Klöfkorn, R. and Nolte, M. and Ohlberger, M.} ,
title = {A Generic Interface for Parallel and Adaptive Scientific Computing: Abstraction Principles and the DUNE-FEM Module} ,
journal = {Computing} ,
year = {2010} ,
volume = {90} ,
number = {3--4} ,
pages = {165--196} ,
url = {http://www.springerlink.com/content/vj103u6079861001/} }

@misc {KoRoeSi:10,
author = {Kristina Kohls and Arnd Rösch and Kunibert G. Siebert} ,
title = {Analysis of Adaptive Finite Elements for Constrained Optimal Control Problems} ,
year = {2010} ,
pages = {308-311} ,
url = {http://dx.doi.org/10.4171/OWR/2010/07} ,
doi = {10.4171/OWR/2010/07} }

@inproceedings {hyp:08,
author = {Dedner, A. and Klöfkorn, R.} ,
editor = {E. Tadmor et al.} ,
title = {Stabilization for Discontinuous Galerkin Methods Applied to Systems of Conservation Laws} ,
booktitle = {Proc. of the 12th International Conference on Hyperbolic Problems, Proceedings of Symposia in Applied Mathematics 67, Part 1, 253-268} ,
year = {2009} }

@phdthesis {kloefkorn:phd,
author = {Klöfkorn, R.} ,
title = {Numerics for Evolution Equations --- A General Interface Based Design Concept} ,
school = {Albert-Ludwigs-Universität Freiburg} ,
year = {2009} ,
note = {http://www.freidok.uni-freiburg.de/volltexte/7175/} }

@incollection {NoSiVe:09,
author = {Nochetto, Ricardo H. and Siebert, Kunibert G. and Veeser, Andreas} ,
editor = {Ronald A. DeVore and Angela Kunoth} ,
title = {Theory of Adaptive Finite Element Methods: An Introduction} ,
booktitle = {Multiscale, Nonlinear and Adaptive Approximation} ,
publisher = {Springer} ,
year = {2009} ,
pages = {409-542} ,
url = {http://dx.doi.org/10.1007/978-3-642-03413-8_12} ,
doi = {10.1007/978-3-642-03413-8_12} }

Abstract: This is a survey on the theory of adaptive finite element methods (AFEM), which are fundamental in modern computational science and engineering. We present a self-contained and up-to-date discussion of AFEM for linear second order elliptic partial differential equations (PDEs) and dimension d>1, with emphasis on the differences and advantages of AFEM over standard FEM. The material is organized in chapters with problems that extend and complement the theory. We start with the functional framework, inf-sup theory, and Petrov-Galerkin method, which are the basis of FEM. We next address four topics of essence in the theory of AFEM that cannot be found in one single article: mesh refinement by bisection, piecewise polynomial approximation in graded meshes, a posteriori error analysis, and convergence and optimal decay rates of AFEM. The first topic is of geometric and combinatorial nature, and describes bisection as a rather simple and efficient technique to create conforming graded meshes with optimal complexity. The second topic explores the potentials of FEM to compensate singular behavior with local resolution and so reach optimal error decay. This theory, although insightful, is insufficient to deal with PDEs since it relies on knowing the exact solution. The third topic provides the missing link, namely a posteriori error estimators, which hinge exclusively on accessible data: we restrict ourselves to the simplest residual-type estimators and present a complete discussion of upper and lower bounds, along with the concept of oscillation and its critical role. The fourth topic refers to the convergence of adaptive loops and its comparison with quasi-uniform refinement. We first show, under rather modest assumptions on the problem class and AFEM, convergence in the natural norm associated to the variational formulation. We next restrict the problem class to coercive symmetric bilinear forms, and show that AFEM is a contraction for a suitable error notion involving the induced energy norm. This property is then instrumental to prove optimal cardinality of AFEM for a class of singular functions, for which the standard FEM is suboptimal.

@inproceedings {AnGaHoKoSiWi:08,
author = {Antil, Harbir and Gantner, Andreas and Hoppe, Ronald H. W. and Köster, Daniel and Siebert, Kunibert G. and Wixforth, Achim} ,
editor = {Langer et al.} ,
title = {Modeling and Simulation of Piezoelectrically Agitated Acoustic Streaming on Microfluidic Biochips.} ,
booktitle = {Domain Decomposition Methods in Science and Engineering XVII} ,
publisher = {Springer} ,
year = {2008} ,
volume = {60} ,
pages = {305-312} ,
url = {http://dx.doi.org/10.1007/978-3-540-75199-1_36} ,
doi = {10.1007/978-3-540-75199-1_36} }

Abstract: Biochips, of the microarray type, are fast becoming the default tool for combinatorial chemical and biological analysis in environmental and medical studies. Programmable biochips are miniaturized biochemical labs that are physically and/or electronically controllable. The technology combines digital photolithography, microfluidics and chemistry. The precise positioning of the samples (e.g., DNA solutes or proteins) on the surface of the chip in pico liter to nano liter volumes can be done either by means of external forces (active devices) or by specific geometric patterns (passive devices). The active devices which will be considered here are nano liter fluidic biochips where the core of the technology are nano pumps featuring surface acoustic waves generated by electric pulses of high frequency. These waves propagate like a miniaturized earthquake, enter the fluid filled channels on top of the chip and cause an acoustic streaming in the fluid which provides the transport of the samples. The mathematical model represents a multiphysics problem consisting of the piezoelectric equations coupled with multiscale compressible Navier-Stokes equations that have to be treated by an appropriate homogenization approach. We discuss the modeling approach and present algorithmic tools for numerical simulations as well as visualizations of simulation results.

@article {dunepaperII:08,
author = {Bastian, P. and Blatt, M. and Dedner, A. and Engwer, C. and Klöfkorn, R. and Kornhuber, R. and Ohlberger, M. and Sander, O.} ,
title = {A Generic Grid Interface for Parallel and Adaptive Scientific Computing. Part II: Implementation and Tests in DUNE} ,
journal = {Computing} ,
year = {2008} ,
volume = {82} ,
number = {2--3} ,
pages = {121--138} ,
url = {http://www.springerlink.com/content/gn177r643q2168g7/} }

@article {dunepaperI:08,
author = {Bastian, P. and Blatt, M. and Dedner, A. and Engwer, C. and Klöfkorn, R. and Ohlberger, M. and Sander, O.} ,
title = {A Generic Grid Interface for Parallel and Adaptive Scientific Computing. Part I: Abstract Framework} ,
journal = {Computing} ,
year = {2008} ,
volume = {82} ,
number = {2--3} ,
pages = {103--119} ,
url = {http://www.springerlink.com/content/4v77662363u41534/} }

@article {CaKrNoSi:08,
author = {J. Manuel Cascón and Christian Kreuzer and Ricardo H. Nochetto and Kunibert G. Siebert} ,
title = {Quasi-Optimal Convergence Rate for an Adaptive Finite Element Method} ,
journal = {SIAM Journal on Numerical Analysis} ,
year = {2008} ,
volume = {46} ,
number = {5} ,
pages = {2524-2550} ,
url = {http://dx.doi.org/10.1137/07069047X} ,
doi = {10.1137/07069047X} }

Abstract: We analyze the simplest and most standard adaptive finite ele- ment method (AFEM), with any polynomial degree, for general second order linear, symmetric elliptic operators. As it is customary in practice, AFEM marks exclusively according to the error estimator and performs a minimal element refinement without the interior node property. We prove that AFEM is a contraction for the sum of energy error and scaled error estimator, be- tween two consecutive adaptive loops. This geometric decay is instrumental to derive optimal cardinality of AFEM. We show that AFEM yields a decay rate of energy error plus oscillation in terms of number of degrees of freedom as dictated by the best approximation for this combined nonlinear quantity.

@incollection {fvca:08,
author = {Dedner, A. and Klöfkorn, R.} ,
title = {The compact discontinuous Galerkin method for elliptic problems} ,
booktitle = {Finite volumes for complex applications V} ,
publisher = {ISTE, London} ,
year = {2008} ,
pages = {761--776} }

@article {comptech:08,
author = {Dedner, A. and Klöfkorn, R. and Kröner,~D.} ,
title = {Efficient higher order methods for convection dominated problems on unstructured grids and applications} ,
journal = {Comput.~Tech.} ,
year = {2008} ,
volume = {13} ,
pages = {25-35} }

@article {heinecj2008,
author = {Heine, C.-J.} ,
title = {Finite element methods on unfitted meshes} ,
journal = {Preprint Fak. f. Math. Phys. Univ. Freiburg} ,
year = {2008} ,
number = {08-09} }

@article {KoKrSi:08,
author = {Daniel Köster and Oliver Kriessl and Kunibert G. Siebert} ,
title = {Design of Finite Element Tools for Coupled Surface and Volume Meshes} ,
journal = {Numerical Mathematics: Theory, Methods and Applications} ,
year = {2008} ,
volume = {1} ,
number = {3} ,
pages = {245-274} ,
url = {http://www.global-sci.org/nmtma/} }

Abstract: Many problems with underlying variational structure involve a coupling of volume with surface effects. A straight-forward approach in a finite element discretization is to make use of the surface triangulation that is naturally induced by the volume triangulation. In an adaptive method one wants to facilitate "matching" local mesh modifications, i.e., local refinement and/or coarsening, of volume and surface mesh with standard tools such that the surface grid is always induced by the volume grid. We describe the concepts behind this approach for bisectional refinement and describe new tools incorporated in the finite element toolbox ALBERTA. We also present several important applications of the mesh coupling.

@incollection {fuel3d:08,
author = {Klöfkorn, R. and Kröner, D. and Ohlberger, M.} ,
editor = {Krebs, H.-J. and Jäger, W.} ,
title = {Parallel Adaptive Simulation of PEM Fuel Cells} ,
booktitle = {Mathematics -- Key Technology for the Future} ,
publisher = {Springer} ,
year = {2008} ,
pages = {235-249} ,
url = {http://dx.doi.org/10.1007/978-3-540-77203-3_16} ,
doi = {10.1007/978-3-540-77203-3_16} }

@article {MoSiVe:08,
author = {Pedro Morin and Kunibert G. Siebert and Andreas Veeser} ,
title = {A Basic Convergence Result for Conforming Adaptive Finite Elements} ,
journal = {Mathematical Models and Methods in Applied Science} ,
year = {2008} ,
volume = {18} ,
pages = {707-737} ,
url = {http://dx.doi.org/10.1142/S0218202508002838} ,
doi = {10.1142/S0218202508002838} }

Abstract: We consider the approximate solution with adaptive finite elements of a class of linear boundary value problems, which includes problems of 'saddle point' type. For the adaptive algorithm we suppose the following framework: refinement relies on unique quasi-regular element subdivisions and generates locally quasi-uniform grids, the finite element spaces are conforming, nested, and satisfy the inf-sup conditions, the error estimator is reliable as well as locally and discretely efficient, and marked elements are subdivided at least once. Under these assumptions, we give a sufficient and essentially necessary condition on marking for the convergence of the finite element solutions to the exact one. This condition is not only satisfied by Dörfler's strategy, but also by the maximum strategy and the equidistribution strategy.

@misc {CaKrNoSi:07,
author = {J. Manuel Cascón and Christian Kreuzer and Ricardo H. Nochetto and Kunibert G. Siebert} ,
title = {Optimal Cardinality of an Adaptive Finite Element Method} ,
year = {2007} ,
pages = {1719-1722} ,
url = {http://dx.doi.org/10.4171/OWR/2007/29} ,
doi = {10.4171/OWR/2007/29} }

@article {CaNoSi:07,
author = {J. Manuel Cascón and Ricardo H. Nochetto and Kunibert G. Siebert} ,
title = {Design and Convergence of AFEM in $H(rm div)$} ,
journal = {Mathematical Models & Methods in Applied Sciences} ,
year = {2007} ,
volume = {17} ,
number = {11} ,
pages = {1849--1881} ,
url = {http://dx.doi.org/10.1142/S0218202507002492} ,
doi = {10.1142/S0218202507002492} }

Abstract: We design an adaptive finite element method (AFEM) for mixed boundary value problems associated with the differential operator A-∇div in H(div, Ω). For A being a variable coefficient matrix with possible jump discontinuities, we provide a complete a posteriori error analysis which applies to both Raviart–Thomas ℝ

@article {GaHoKoeSiWi:07,
author = {Andreas Ganter and Ronald H. W. Hoppe and Daniel Köster and Kunibert G. Siebert and Achim Wixforth} ,
title = {Numerical Simulation of Piezoelectrically Agitated Surface Acoustic Waves on Microfluidic Biochips} ,
journal = {Computing and Visualization in Science} ,
year = {2007} ,
volume = {10} ,
number = {3} ,
pages = {145-161} ,
url = {http://dx.doi.org/10.1007/s00791-006-0040-y} ,
doi = {10.1007/s00791-006-0040-y} }

Abstract: Microfluidic biochips are biochemical laboratories on the microscale that are used for genotyping and sequencing in genomics, protein profiling in proteomics, and cytometry in cell analysis. There are basically two classes of such biochips: active devices, where the solute transport on a network of channels on the chip surface is realized by external forces, and passive chips, where this is done using a specific design of the geometry of the channel network. Among the active biochips, current interest focuses on devices whose operational principle is based on piezoelectrically driven surface acoustic waves (SAWs) generated by interdigital transducers placed on the chip surface. In this paper, we are concerned with the numerical simulation of such piezoelectrically agitated SAWs relying on a mathematical model that describes the coupling of the underlying piezoelectric and elastomechanical phenomena. Since the interdigital transducers usually operate at a fixed frequency, we focus on the time-harmonic case. Its variational formulation gives rise to a generalized saddle point problem for which a Fredholm alternative is shown to hold true. The discretization of the time-harmonic surface acoustic wave equations is taken care of by continuous, piecewise polynomial finite elements with respect to a nested hierarchy of simplicial triangulations of the computational domain. The resulting algebraic saddle point problems are solved by blockdiagonally preconditioned iterative solvers with preconditioners of BPX-type. Numerical results are given both for a test problem documenting the performance of the iterative solution process and for a realistic SAW device illustrating the properties of SAW propagation on piezoelectric materials.

@article {MoSiVe:07d,
author = {Pedro Morin and Kunibert G. Siebert and Andreas Veeser} ,
title = {Basic Convergence Results for Conforming Adaptive Finite Elements} ,
journal = {Proceedings in Applied Mathematics and Mechanics} ,
year = {2007} ,
volume = {7} ,
number = {1} ,
pages = {1026001-1026002} ,
note = {Special Issue: Sixth International Congress on Industrial Applied Mathematics (ICIAM07) and GAMM Annual Meeting, Zürich 2007} ,
url = {http://dx.doi.org/10.1002/pamm.200700771} ,
doi = {10.1002/pamm.200700771} }

Abstract: We report on a result establishing plain convergence for conforming adaptive finite elements under relatively general assumptions on problem class and adaptive algorithm. Moreover, we indicate some applications and give a sketch of the proof. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

@misc {MoSiVe:07a,
author = {Pedro Morin and Kunibert G. Siebert and Andreas Veeser} ,
title = {A Basic Convergence Result for Conforming Adaptive Finite Element Methods} ,
year = {2007} ,
pages = {1705-1708} ,
url = {http://dx.doi.org/10.4171/OWR/2007/29} ,
doi = {10.4171/OWR/2007/29} }

@inproceedings {MoSiVe:07b,
author = {Pedro Morin and Kunibert G. Siebert and Andreas Veeser} ,
editor = {V. Cutello and G. Fotia and L. Puccio} ,
title = {Convergence of Finite Elements Adapted for Weaker Norms} ,
booktitle = {Applied and Industrial Matematics in Italy - II} ,
publisher = {World Sci. Publ.} ,
year = {2007} ,
volume = {75} ,
pages = {468-479} ,
url = {http://dx.doi.org/10.1142/9789812709394_0041} ,
doi = {10.1142/9789812709394_0041} }

Abstract: We consider finite elements that are adapted to a (semi)norm that is weaker than the one of the trial space. We establish convergence of the finite element solutions to the exact one under the following conditions: refinement relies on unique quasi-regular element subdivisions and generates locally quasi-uniform grids; the finite element spaces are conforming, nested, and satisfy the inf-sup condition; the error estimator is reliable and appropriately locally efficient; the indicator of a non-marked element is bounded by the estimator contribution associated with the marked elements, and each marked element is subdivided at least once. This abstract convergence result is illustrated by two examples.

@article {SiebertVeeser:07,
author = {Siebert, Kunibert G. and Veeser, Andreas} ,
title = {A Unilaterally Constrained Quadratic Minimization with Adaptive Finite Elements} ,
journal = {SIAM Journal on Optimization} ,
year = {2007} ,
volume = {18} ,
number = {1} ,
pages = {260-289} ,
url = {http://dx.doi.org/10.1137/05064597X} ,
doi = {10.1137/05064597X} }

Abstract: We consider obstacle problems where a quadratic functional associated with the Laplacian is minimized in the set of functions above a possibly discontinuous and thin but piecewise affine obstacle. In order to approximate minimum point and value, we propose an adaptive algorithm that relies on minima with respect to admissible linear finite element functions and on an a posteriori estimator for the error in the minimum value. It is proven that the generated sequence of approximate minima converges to the exact one. Furthermore, our numerical results in two and three dimensions indicate that the convergence rate with respect to the number of degrees of freedom is optimal in that it coincides with the one of nonlinear or adaptive approximation.

@article {bfg:06,
author = {Backofen, R. and Borrmann, H.-G. and Deck, W. and Dedner, A. and De Raedt, L. and Desch, K. and Diesmann, M. and Geier, M. and Greiner, A. and Hess, W.R. and Honerkamp, J. and Jankowski, S. and Krossing, I. and Liehr, A.W. and Karwathi, A. and Klöfkorn, R. and Pesché, R. and Potjans, T.C. and Röttger, M.C. and Schmidt-Thieme, L. and Schneider, G. and Voß, B. and Wiebelt, B. and Wienemann, P. and Winterer, V.-H.} ,
title = {A Bottom-up approach to Grid-Computing at a University: the Black-Forest-Grid Initiative} ,
journal = {Praxis der Informationsverarbeitung und Kommunikation} ,
year = {2006} ,
volume = {29} ,
number = {2} ,
pages = {81-87} }

@inproceedings {dune:06,
author = {Bastian, P. and Blatt, M. and Dedner, A. and Engwer, Ch. and Klöfkorn, R. and Kuttanikkad, S.P. and Ohlberger, M. and Sander, O.} ,
title = {The Distributed and Unified Numerics Environment (DUNE)} ,
booktitle = {Proc. of the 19th Symposium on Simulation Technique in Hannover, Sep. 12 - 14} ,
year = {2006} }

@incollection {bddko:06,
author = {Burri, A. and Dedner, A. and Diehl, D. and Klöfkorn, R. and Ohlberger, M.} ,
editor = {Y. Shokin et al.} ,
title = {A general object oriented framework for discretizing non-linear evolution equations} ,
booktitle = {Advances in High Performance Computing and Computational Sciences} ,
publisher = {Springer} ,
year = {2006} ,
volume = {93} ,
pages = {69-87} ,
url = {http://dx.doi.org/10.1007/978-3-540-33844-4_7} ,
doi = {10.1007/978-3-540-33844-4_7} }

@incollection {dunepara:05,
author = {Burri, A. and Dedner, A. and Klöfkorn, R. and Ohlberger, M.} ,
editor = {E. Krause et al.} ,
title = {An efficient implementation of an adaptive and parallel grid in DUNE} ,
booktitle = {Computational Science and High Performance Computing II} ,
publisher = {Springer} ,
year = {2006} ,
volume = {91} ,
pages = {67-82} ,
url = {http://dx.doi.org/10.1007/3-540-31768-6_7} ,
doi = {10.1007/3-540-31768-6_7} }

@article {heinecj2006,
author = {Heine, C.-J.} ,
title = {Computations of form and stability of rotating drops with finite elements} ,
journal = {IMA J. Numer. Anal.} ,
year = {2006} ,
volume = {26} ,
number = {4} ,
pages = {723--751} }

@article {NoSchSiVe:06,
author = {Nochetto, Ricardo H. and Schmidt, Alfred and Siebert, Kunibert G. and Veeser, Andreas} ,
title = {Pointwise A Posteriori Error Estimates for Monotone Semi-linear Equations} ,
journal = {Numerische Mathematik} ,
year = {2006} ,
volume = {104} ,
number = {4} ,
pages = {515-538} ,
url = {http://dx.doi.org/10.1007/s00211-006-0027-0} ,
doi = {10.1007/s00211-006-0027-0} }

Abstract: We derive upper and lower a posteriori estimates for the maximum norm error in finite element solutions of monotone semi-linear equations. The estimates hold for Lagrange elements of any fixed order, non-smooth nonlinearities, and take numerical integration into account. The proof hinges on constructing continuous barrier functions by correcting the discrete solution appropriately, and then applying the continuous maximum principle; no geometric mesh constraints are thus required. Numerical experiments illustrate reliability and efficiency properties of the corresponding estimators and investigate the performance of the resulting adaptive algorithms in terms of the polynomial order and quadrature.

@inproceedings {dune:04,
author = {Bastian, P. and Droske, M. and Engwer, Ch. and Klöfkorn, R. and Neubauer, T. and Ohlberger, M. and Rumpf, M.} ,
editor = {T. Barth et al.} ,
title = {Towards a Unified Framework for Scientific Computing} ,
booktitle = {Domain Decomposition Methods in Science and Engineering} ,
publisher = {Springer} ,
year = {2005} ,
number = {40} ,
pages = {167-174} ,
url = {http://dx.doi.org/10.1007/3-540-26825-1_13} ,
doi = {10.1007/3-540-26825-1_13} }

@article {NoSiVe:05,
author = {Nochetto, Ricardo H. and Siebert, Kunibert G. and Veeser, Andreas} ,
title = {Fully Localized A Posteriori Error Estimators and Barrier Sets for Contact Problems} ,
journal = {SIAM Journal on Numerical Analysis} ,
year = {2005} ,
volume = {42} ,
number = {5} ,
pages = {2118-2135} ,
url = {http://dx.doi.org/10.1137/S0036142903424404} ,
doi = {10.1137/S0036142903424404} }

Abstract: We derive novel pointwise a posteriori error estimators for elliptic obstacle problems which, except for obstacle resolution, completely vanish within the full-contact set (localization). We then construct a posteriori barrier sets for free boundaries under a natural stability (or nondegeneracy) condition. We illustrate localization properties as well as reliability and efficiency for both solutions and free boundaries via several simulations in 2 and 3 dimensions.

@book {SchmidtSiebert:05,
author = {Alfred Schmidt and Kunibert G. Siebert} ,
editor = {T.J. Barth, M. Griebel, D.E. Keyes, R.M. Nieminen, D. Roose, T.Schlick} ,
title = {Design of Adaptive Finite Element Software. The Finite Element Toolbox ALBERTA} ,
publisher = {Springer} ,
year = {2005} ,
volume = {42} ,
url = {http://www.alberta-fem.de} ,
doi = {10.1007/b138692} }

Abstract: During the last years, scientific computing has become an important research branch located between applied mathematics and applied sciences and engineering. Highly efficient numerical methods are based on adaptive methods, higher order discretizations, fast linear and non-linear iterative solvers, multi-level algorithms, etc. Such methods are integrated in the adaptive finite element software ALBERTA. It is a toolbox for the fast and flexible implementation of efficient software for real life applications, based on modern algorithms. ALBERTA also serves as an environment for improving existent, or developing new numerical methods in an interplay with mathematical analysis and it allows the direct integration of such new or improved methods in existing simulation software. The book is accompanied by a full distribution of ALBERTA (Version 1.2) on a CD including an implementation of several model problems. System requirements for ALBERTA are a Unix/Linux environment with C and FORTRAN Compilers, OpenGL graphics and GNU make. These model implementations serve as a basis for students and researchers for the implementation of their own research projects within ALBERTA.

@misc {SiebertVeeser:05,
author = {Kunibert G. Siebert and Andreas Veeser} ,
title = {Convergence of the Equidistribution Strategy} ,
year = {2005} ,
pages = {2129-2131} ,
url = {http://dx.doi.org/10.4171/OWR/2005/37} ,
doi = {10.4171/OWR/2005/37} }

@article {BaBaSie:04,
author = {Bamberger, Andreas and Bänsch, Eberhard and Siebert, Kunibert G.} ,
title = {Experimental and numerical investigation of edge tones} ,
journal = {ZAMM Journal of Applied Mathematics and Mechanics} ,
year = {2004} ,
volume = {84} ,
number = {9} ,
pages = {632-646} ,
url = {http://dx.doi.org/10.1002/zamm.200310122} ,
doi = {10.1002/zamm.200310122} }

Abstract: We study both, by experimental and numerical means the fluid dynamical phenomenon of edge tones. Of particular interest is the verification of scaling laws relating the frequency f to given quantities, namely d, the height of the jet, w, the stand–off distance and the velocity of the jet. We conclude that the Strouhal number Sd is related to the geometrical quantities through Sd = C ⋅ (d / w)n with n ≈ 1, in contrast to some analytical treatments of the problem. The constant C of the experiment agrees within 13–15% with the result of the numerical treatment. Only a weak dependence on the Reynolds number with respect to d is observed. In general, a very good agreement of the experimental and the numerical simulations is found.

@article {heinecj2004,
author = {Heine, C.-J.} ,
title = {Isoparametric finite element approximation of curvature on hypersurfaces} ,
journal = {Preprint Fak. f. Math. Phys. Univ. Freiburg} ,
year = {2004} ,
number = {26} }

@misc {BoSchSiBaDzBeKa:03,
author = {Boschert, Stefan and Schmidt, Alfred and Siebert, Kunibert G. and Bänsch, Eberhard and Dziuk, Gerhard and Benz, Klaus-Werner and Kaiser, Thomas} ,
title = {Simulation of Industrial Crystal Growth by the Vertical Bridgman Method} ,
year = {2003} }

Abstract: Single crystals of Cadmium-Zinc-Telluride are used as a substrate material for the production of infrared detectors and are usually grown by the vertical Bridgman method. We present a simulation of the whole growth process in two steps: In the first step, the (stationary) heat transport in the furnace is modeled and calculated for different positions of the ampoule. This provides information about the most important parameter during this process: the temperature distribution in furnace and ampoule. The obtained temperatures are then used in the second step as boundary conditions for the (time dependent) simulation of temperature and convection in the ampoule. Only the use of adaptive finite element methods allows an efficient numerical simulation of the moving phase boundary, the convection in the melt and the temperature distribution in melt and crystal. Numerical results are presented for both furnace and ampoule simulations.

@inproceedings {DoerflerSiebert:03,
author = {Dörfler, Willy and Siebert, Kunibert G.} ,
editor = {S. Hildebrandt, H. Karcher} ,
title = {An Adaptive Finite Element Method for Minimal Surfaces} ,
booktitle = {Geometric Analysis and Nonlinear Partial Differential Equations} ,
publisher = {Springer} ,
year = {2003} ,
pages = {146-175} ,
url = {http://www.springer.com/mathematics/dynamical+systems/book/978-3-540-44051-2} }

@article {HORSS03,
author = {Haasdonk, Bernard and Ohlberger, Mario and Rumpf, Martin and Schmidt, Alfred and Siebert, Kunibert G.} ,
title = {Multiresolution Visualization of Higher Order Adaptive Finite Element Simulations} ,
journal = {Computing} ,
year = {2003} ,
volume = {70} ,
number = {3} ,
pages = {181-204} ,
doi = {10.1007/s00607-003-1476-2} }

Abstract: h–p–adaptive projection with respect to any prescribed threshold value for the visual error. This projection can then be processed by various local rendering methods, e.g. color coding of data or isosurface extraction. Especially for color coding purposes modern texture capabilities are used to directly render higher polynomial data by superposition of polynomial basis function textures and final color look-up tables. Numerical experiments from CFD clearly demonstrate the applicability and efficiency of our approach.

@techreport {fuel:03,
author = {Kühn, K. and Ohlberger, M. and Schumacher, J.O. and Ziegler, C. and Klöfkorn, R.} ,
title = {A dynamic two-phase flow model of proton exchange membrane fuel cells} ,
year = {2003} ,
note = {presented at the 2nd EUROPEAN PEM FUEL CELL FORUM, Lucerne} }

@article {MoNoSi:03,
author = {Pedro Morin and Ricardo H. Nochetto and Kunibert G. Siebert} ,
title = {Local Problems on Stars: A Posteriori Error Estimators, Convergence, and Performance} ,
journal = {Mathematics of Computation} ,
year = {2003} ,
volume = {72} ,
number = {243} ,
pages = {1067-1097} ,
url = {http://dx.doi.org/10.1090/S0025-5718-02-01463-1} ,
doi = {10.1090/S0025-5718-02-01463-1} }

Abstract: A new computable a posteriori error estimator is introduced, which relies on the solution of small discrete problems on stars. It exhibits built-in flux equilibration and is equivalent to the energy error up to data oscillation without any saturation assumption. A simple adaptive strategy is designed, which simultaneously reduces error and data oscillation, and is shown to converge without mesh pre-adaptation nor explicit knowledge of constants. Numerical experiments reveal a competitive performance, show extremely good effectivity indices, and yield quasi-optimal meshes.

@article {NoSiVe:03,
author = {Nochetto, Ricardo H. and Siebert, Kunibert G. and Veeser, Andreas} ,
title = {Pointwise A Posteriori Error Control for Elliptic Obstacle Problems} ,
journal = {Numerische Mathematik} ,
year = {2003} ,
volume = {95} ,
number = {1} ,
pages = {163-195} ,
url = {http://dx.doi.org/10.1007/s00211-002-0411-3} ,
doi = {10.1007/s00211-002-0411-3} }

Abstract: We consider a finite element method for the elliptic obstacle problem over polyhedral domains in R^d , which enforces the unilateral constraint solely at the nodes. We derive novel optimal upper and lower a posteriori error bounds in the maximum norm irrespective of mesh fineness and the regularity of the obstacle, which is just assumed to be Hölder continuous. They exhibit optimal order and localization to the non-contact set. We illustrate these results with simulations in 2d and 3d showing the impact of localization in mesh grading within the contact set along with quasi-optimal meshes.

@article {diplpaper:02,
author = { Klöfkorn, R. and Kröner, D. and Ohlberger, M.} ,
title = {Local adaptive methods for convection dominated problems.} ,
journal = {Int. J. Numer. Methods Fluids} ,
year = {2002} ,
volume = {40} ,
number = {1-2} ,
pages = {79-91} ,
url = {http://onlinelibrary.wiley.com/doi/10.1002/fld.268/abstract} }

@article {LiBoDoBeKrSchSiDz:02,
author = {K.-M. Lin and S. Boschert and P. Dold and K. W. Benz and O. Kriessl and A. Schmidt and K. G. Siebert and G. Dziuk} ,
title = {Numerical Methods for Industrial Bridgman Growth of (Cd,Zn)Te} ,
journal = {Journal of Crystal Growth} ,
year = {2002} ,
volume = {237-239} ,
pages = {1736-1740} ,
url = {http://dx.doi.org/10.1016/S0022-0248(01)02321-1} ,
doi = {10.1016/S0022-0248(01)02321-1} }

Abstract: This paper presents efficient numerical methods—the “inverse modeling” method and the adaptive finite element method—for optimizing the heat transport as well as for investigating the heat and mass transport under the influence of convection during crystal growth, especially near the liquid/solid interface. These methods have been applied to industrial Bridgman-furnaces for the growth of 65–75 mm diameter (Cd,Zn)Te crystals.

@article {MoNoSi:02,
author = {Pedro Morin and Ricardo H. Nochetto and Kunibert G. Siebert} ,
title = {Convergence of Adaptive Finite Element Methods} ,
journal = {SIAM Review} ,
year = {2002} ,
volume = {44} ,
number = {4} ,
pages = {631-658} ,
url = {http://dx.doi.org/10.1016/j.apnum.2008.12.006} ,
doi = {10.1016/j.apnum.2008.12.006} }

Abstract: The a priori convergence of finite element methods is based on the density property of the sequence of finite element spaces which essentially assumes a quasi-uniform mesh-refining. The advantage is guaranteed convergence for a large class of data and solutions; the disadvantage is a global mesh refinement everywhere accompanied by large computational costs.

Adaptive finite element methods (AFEMs) automatically refine exclusively wherever their refinement indication suggests to do so and consequently leave out refinements at other locations. In other words, the density property is violated on purpose and the a priori convergence is not guaranteed automatically and, in fact, crucially depends on algorithmic details. The advantage of AFEMs is a more effective mesh in many practical examples accompanied by smaller computational costs; the disadvantage is that the desirable convergence property is not guaranteed a priori. Efficient error estimators can justify a numerical approximation a posteriori and so achieve reliability. But it is not theoretically justified from the start that the adaptive mesh-refinement will generate an accurate solution at all.

In order to foster the development of a convergence theory and improved design of AFEMs in computational engineering and sciences, this paper describes a particular version of an AFEM and analyses convergence results for three model problems in computational mechanics: linear elastic material (A), nonlinear monotone elastic material (B), and Hencky elastoplastic material (C). It establishes conditions sufficient for error-reduction in (A), for energy-reduction in (B), and eventually for strong convergence of the stress field in (C) in the presence of small hardening.

@phdthesis {kloefkorn:dipl,
author = {Klöfkorn, R.} ,
title = {Simulation von Abbau- und Transportprozessen gelöster Schadstoffe im Grundwasser} ,
school = {Albert-Ludwigs-Universität Freiburg} ,
year = {2001} }

@article {SchmidtSiebert:01,
author = {Alfred Schmidt and Kunibert G. Siebert} ,
title = {ALBERT --- Software for Scientific Computations and Applications} ,
journal = {Acta Mathematica Universitatis Comenianae, New Ser.} ,
year = {2001} ,
volume = {70} ,
number = {1} ,
pages = {105-122} ,
url = {http://www.emis.de/journals/AMUC/_vol-70/_no_1/_siebert/siebert.html} }

Abstract: Adaptive finite element methods are a modern, widely used tool which make realistic computations feasible, even in three space dimensions. We describe the basic ideas and ingredients of adaptive FEM and the implementation of our toolbox The design of ALBERT is based on the natural hierarchy of locally refined meshes and an abstract concept of general finite element spaces. As a result, dimension independent programming of applications is possible. Numerical results from applications in two and three space dimensions demonstrate the flexibility of

@incollection {BoSchSi:00,
author = {Stefan Boschert and Alfred Schmidt and Kunibert G. Siebert} ,
editor = {J. S. Szmyd and K. Suzuki} ,
title = {Numerical Simulation of Crystal Growth by the Vertical Bridgman Method} ,
booktitle = {Modelling of Transport Phenomena in Crystal Growth} ,
publisher = {WIT Press} ,
year = {2000} ,
volume = {6} ,
pages = {315-330} ,
url = {http://www.witpress.com/978-1-85312-735-9.html} }

@article {DeckelnickSiebert:00,
author = {Deckelnick, Klaus and Siebert, Kunibert G.} ,
title = {$W^1,infty$-Convergence of the Discrete Free Boundary for Obstacle Problems} ,
journal = {IMA Journal of Numerical Analysis} ,
year = {2000} ,
volume = {20} ,
number = {3} ,
pages = {481-498} ,
url = {http://dx.doi.org/10.1093/imanum/20.3.481} ,
doi = {10.1093/imanum/20.3.481} }

Abstract: We examine the discrete free boundaries arising from a finite element discretization of a variational inequality. We give L∞ error bounds for the Hausdorff distance of the discrete and true free boundary, as well as for the normals. The theoretical results are confirmed by numerical experiments in two and three dimensions.

@article {MoNoSi:00,
author = {Pedro Morin and Ricardo H. Nochetto and Kunibert G. Siebert} ,
title = {Data Oscillation and Convergence of Adaptive FEM} ,
journal = {SIAM Journal on Numerical Analysis} ,
year = {2000} ,
volume = {38} ,
number = {2} ,
pages = {466-488} ,
url = {http://dx.doi.org/10.1137/S0036142999360044} ,
doi = {10.1137/S0036142999360044} }

Abstract: Data oscillation is intrinsic information missed by the averaging process associated with finite element methods (FEM) regardless of quadrature. Ensuring a reduction rate of data oscillation, together with an error reduction based on a posteriori error estimators, we construct a simple and efficient adaptive FEM for elliptic PDE with linear rate of convergence without any preliminary mesh adaptation nor explicit knowledge of constants. Any prescribed error tolerance is thus achieved in a finite number of steps. A number of numerical experiments in 2d and 3d yield quasi-optimal meshes along with a competitive performance. Key words. A posteriori error estimators, data oscillation, adaptive mesh refinement, convergence, performance, quasi-optimal meshes 1991 AMS subject classification. 65N12, 65N15, 65N30, 65N50, 65Y20 1 Introduction and Main Results Adaptive procedures for the numerical solution of partial differential equations (PDE) started in the late 70's and are now standard tools...

@article {SchmidtSiebert:00,
author = {Schmidt, Alfred and Siebert, Kunibert G.} ,
title = {A Posteriori Estimators for the $h$-$p$ Version of the Finite Element Method in 1d} ,
journal = {Applied Numerical Mathematics} ,
year = {2000} ,
volume = {35} ,
number = {1} ,
pages = {43-66} ,
url = {http://dx.doi.org/10.1016/S0168-9274(99)00046-X} ,
doi = {10.1016/S0168-9274(99)00046-X} }

Abstract: We consider the h – p finite element method for elliptic problems in one dimension. The strategy for choosing an h - or p -enrichment for an element which is subject to a refinement in an adaptive method is not well understood; in particular, this is the most important open problem associated with h – p -refinement. A mathematical derivation of a posteriori estimates for the error and the error reduction corresponding to an h - or p -refinement of elements is presented. The estimation of the error reduction uses the solution of local problems, the estimate is bounded by the true reduction from below and above with constants only depending on the differential operator. Based on these a posteriori estimates an adaptive algorithm is derived. Numerical results show the efficiency of the estimators for several problem classes. For the xα model singularity the a priori known optimal h – p mesh is obtained by this algorithm.

@article {SchmidtSiebert:99,
author = {Schmidt, Alfred and Siebert, Kunibert G.} ,
title = {Abstract Data Structures for a Finite Element Package: Design Principles of ALBERT} ,
journal = {Journal of Applied Mathematics and Mechanics} ,
year = {1999} ,
volume = {79} ,
number = {1} ,
pages = {49-52} ,
url = {http://www.alberta-fem.de/design.html} }

Abstract: ALBERTA is an Adaptive multiLevel finite element toolbox using Bisectioning refinement and Error control by Residual Techniques for scientific Applications. Its design is based on appropriate data structures holding geometrical, finite element, and algebraic information. Using such data structures, abstract adaptive methods for stationary and time dependent problems, assembly tools for discrete systems, and dimension dependent tasks like mesh modifications can be provided in a library. This allows dimension-independent development and programming of a general class of applications. In ALBERTA, hierarchical 2d and 3d meshes are stored in binary trees. Several sets of finite elements can be used on the same mesh, either using predefined ones, or by adding new sets for special applications. Depending on the currently used finite element spaces, all degrees of freedom are automatically managed during mesh modifications.

@inproceedings {BKSSBD:98,
author = {Boschert, Stefan and Kaiser, Thomas and Schmidt, Alfred and Siebert, Kunibert G. and Benz, Klaus-Werner and Dziuk, Gerhard} ,
editor = {S.N. Atluri and P.E. O'Donoghue} ,
title = {Global Simulation of (Cd,Zn)Te Single Crystal Growth by the Vertical Bridgman Technique} ,
booktitle = {Modeling and Simulation Based Engineering} ,
publisher = {Tech Science Press} ,
year = {1998} ,
url = {http://www.techscience.com/books/msbe_hc_rm.html} }

@misc {SchmidtSiebert:98,
author = {Schmidt, Alfred and Siebert, Kunibert G.} ,
title = {Concepts of the Finite Element Toolbox ALBERT} ,
year = {1998} }

Abstract: Introduction and design principles The core part of every finite element program is the problem--dependent assembly and solution of the discretized problem. This holds for programs which solve the discrete problem on a fixed mesh as well as for adaptive methods which automatically adjust the underlying mesh to the actual problem and solution. In an adaptive iteration, the solution of a discrete system is necessary after each mesh change. A general finite element toolbox must provide flexibility in problems and finite element spaces while on the other hand this core part can be performed efficiently. Data structures are needed which allow an easy and efficient implementation of the problem--dependent parts and also allow to use adaptive methods, mesh modification algorithms, and solvers for linear and nonlinear discrete problems by calling library routines. Starting point for our considerations is the abstract concept of a finite element space defined (similar to the definition

@misc {Siebert:98,
author = {Siebert, Kunibert G.} ,
title = {Einführung in die numerische Behandlung der Navier-Stokes-Gleichungen} ,
year = {1998} }

Abstract: Ausarbeitung einer Vorlesung zum
Praktikum "Numerik Partieller Differentialgleichungen III"

Wintersemester 1997/98

Abstract:

Das Zwischenschritt-Theta Verfahren zur Zeitdiskretisierung der instationären Navier-Stokes-Gleichungen wird vorgestellt. Wird dieses Verfahren als Operator-Splitting verwendet, so sind in einem Zeitschritt zwei lineare Sattelpunktprobleme und eine nicht-lineare elliptische Gleichung zu lösen. Numerische Analysis und Verfahren zur Lösung von Sattelpunktproblemen werden im Detail vorgestellt.
Inhaltsverzeichnis:

1. Formulierung der Navier-Stokes-Gleichungen
2. Zeit-Diskretisierung
1. Abstraktes Operator-Splitting
2. Konsistenz- und Stabilitätseigenschaften
3. Operator Splitting für Navier--Stokes
3. Lösung des linearen Teilproblems
1. Sattelpunktprobleme
2. Existenz und Eindeutigkeit der Lösungeines Sattelpunktproblems
3. Existenz und Eindeutigkeit des Quasi-Stokes Problems
4. Diskretisierung von Sattelpunktproblemen
5. Stabile Diskretisierungen für Quasi-Stokes
6. Gradienten-Verfahren im Hilbertraum
1. Methode des steilsten Abstiegs
2. CG-Verfahren
7. CG-Verfahren für Sattelpunktprobleme
8. Vorkonditioniertes CG--Verfahren für Quasi-Stokes

@article {RuSchmSie:96,
author = {Rumpf, Martin and Schmidt, Alfred and Siebert, Kunibert G.} ,
title = {Functions Defining Arbitrary Meshes --- A Flexible Interface Between Numerical Data and Visualization Routines} ,
journal = {Computer Graphics Forum} ,
year = {1996} ,
volume = {15} ,
number = {2} ,
pages = {129-141} ,
url = {http://dx.doi.org/10.1111/1467-8659.1520129} ,
doi = {10.1111/1467-8659.1520129} }

@misc {SchmidtSiebert:96,
author = {Schmidt, Alfred and Siebert, Kunibert G.} ,
title = {Numerical Aspects of Parabolic Free Boundary Problems - Adaptive Finite Element Methods.} ,
year = {1996} }

@article {Siebert:96,
author = {Siebert, Kunibert G.} ,
title = {An A Posteriori Error Estimator for Anisotropic Refinement} ,
journal = {Numerische Mathematik} ,
year = {1996} ,
volume = {73} ,
number = {3} ,
pages = {373-398} ,
url = {http://dx.doi.org/10.1007/s002110050197} ,
doi = {10.1007/s002110050197} }

Abstract: Besides an algorithm for local refinement, an a posteriori error estimator is the basic tool of every adaptive finite element method. Using information generated by such an error estimator the refinement of the grid is controlled. For 2nd order elliptic problems we present an error estimator for anisotropically refined grids, like n -d cuboidal and 3-d prismatic grids, that gives correct information about the size of the error; additionally it generates information about the direction into which some element has to be refined to reduce the error in a proper way. Numerical examples are presented for 2-d rectangular and 3-d prismatic grids.

@misc {BaenschSiebert:95,
author = {Bänsch, Eberhard and Siebert, Kunibert G.} ,
title = {A Posteriori Error Estimation for Nonlinear Problems by Duality Techniques} ,
year = {1995} }

Abstract: We present an abstract framework for a posteriori error estimation in finite element methods for a quite general class of nonlinear problems
F(u) = 0 in X*
where X is a Hilbert space, X* its dual and F: X -> X*. The error between exact and discrete solution is estimated in a weaker norm than the corresponding energy norm. Using the Aubin-Nitsche-trick, the error is represented by a linear dual problem. Assuming that the solution of the dual problem is regular, i. e. the solution belongs to a subspace W of X with a stronger norm, the error is estimated by the weaker W*-norm of the residual from above and below.
Since it is not possible to compute the W*-norm of the residual, we also present utilities for an estimation of this norm for second order problems. In second order problems we want to estimate the error in the L^2-norm. For a localization of the residual we have to construct suitable cut-off functions which have weak second derivatives. This reflects the regularity of the dual problem on the discrete level and yields an estimation of the error by the error estimator from above and below.
Concrete error estimators for second order semi-linear and eigenvalue problems are presented
AMS-Classification. 65N30, 65N50
Freiburg, Preprint Nr. 30/1995

@inproceedings {RuSchmSie:95,
author = {Rumpf, Martin and Scmidt, Alfred and Siebert, Kunibert G.} ,
editor = {R. Scateni, J. Van Wijk, P. Zanarini} ,
title = {On a Unified Visualization Approach for Data from Advanced Numerical Methods} ,
booktitle = {Visualization in Scientific Computing '95} ,
publisher = {Springer} ,
year = {1995} ,
pages = {35-44} }

@article {Siebert:93b,
author = {Siebert, Kunibert G.} ,
title = {Local Refinement of 3D-Meshes Consisting of Prisms and Conforming Closure} ,
journal = {IMPACT of Computing in Science and Engineering} ,
year = {1993} ,
volume = {5} ,
number = {4} ,
pages = {271-284} ,
url = {http://dx.doi.org/10.1006/icse.1993.1012} ,
doi = {10.1006/icse.1993.1012} }

Abstract: An algorithm for the local refinement of a given triangulation consisting of prisms is presented. In the refined triangulation there can be some nonconforming nodes. It is shown that there exists a conforming triangulation consisting of prisms, pyramids, and tetrahedra which contains the nonconforming one. Proofs for the finiteness of the algorithm and stability of the obtained triangulations are presented.

@phdthesis {Siebert:93a,
author = {Siebert, Kunibert G.} ,
title = {An A Posteriori Error Estimator for Anisotropic Refinement} ,
school = {Freiburg} ,
year = {1993} }

@mastersthesis {Siebert:90,
author = {Siebert, Kunibert G.} ,
title = {Ein Finite-Elemente-Verfahren zur Lösung der inkompressiblen Euler-Gleichungen auf der Sphäre mit der Stromlinien-Diffusions-Methode} ,
school = {Bonn} ,
year = {1990} }