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37
Tetrahedral Grid Refinement
, 1995
"... Zusammenfassung Tetrahedral Grid Refinement. We present a refinement algorithm for unstructured tetrahedral grids, which generates possibly highly nonuniform but nevertheless consistent (closed) and stable triangulations. Therefore we first define some local regular and irregular refinement rules t ..."
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Cited by 51 (1 self)
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Zusammenfassung Tetrahedral Grid Refinement. We present a refinement algorithm for unstructured tetrahedral grids, which generates possibly highly nonuniform but nevertheless consistent (closed) and stable triangulations. Therefore we first define some local regular and irregular refinement rules that are applied to single elements. The global refinement algorithm then describes how these local rules can be combined and rearranged in order to ensure consistency as well as stability. It is given in a rather general form and includes also grid coarsening. 1991 Mathematics Subject Classifications: 65N50, 65N55 Key words: Tetrahedral grid refinement, stable refinements, consistent triangulations, green closure, grid coarsening. Verfeinerung von TetraederGittern. Es wird ein Verfeinerungsalgorithmus fur unstrukturierte TetraederGitter vorgestellt, der moglicherweise stark nichtuniforme aber dennoch konsistente (d.h. geschlossene) und stabile Triangulierungen liefert. Dazu definieren w...
deal.II – a general purpose object oriented finite element library
 ACM TRANS. MATH. SOFTW
"... An overview of the software design and data abstraction decisions chosen for deal.II, a general purpose finite element library written in C++, is given. The library uses advanced objectoriented and data encapsulation techniques to break finite element implementations into smaller blocks that can be ..."
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Cited by 49 (17 self)
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An overview of the software design and data abstraction decisions chosen for deal.II, a general purpose finite element library written in C++, is given. The library uses advanced objectoriented and data encapsulation techniques to break finite element implementations into smaller blocks that can be arranged to fit users requirements. Through this approach, deal.II supports a large number of different applications covering a wide range of scientific areas, programming methodologies, and applicationspecific algorithms, without imposing a rigid framework into which they have to fit. A judicious use of programming techniques allows to avoid the computational costs frequently associated with abstract objectoriented class libraries. The paper presents a detailed description of the abstractions chosen for defining geometric information of meshes and the handling of degrees of freedom associated with finite element spaces, as well as of linear algebra, input/output capabilities and of interfaces to other software, such as visualization tools. Finally, some results obtained with applications built atop deal.II are shown to demonstrate the powerful capabilities of this toolbox.
An Adaptive Finite Element Method for Large Scale Image Processing
 INTERNATIONAL CONFERENCE ON SCALESPACE THEORIES IN COMPUTER VISION
, 1999
"... Nonlinear diffusion methods have proved to be powerful methods in the processing of 2D and 3D images. They allow a denoising and smoothing of image intensities while retaining and enhancing edges. As time evolves in the corresponding process, a scale of successively coarser image details is generate ..."
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Cited by 38 (18 self)
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Nonlinear diffusion methods have proved to be powerful methods in the processing of 2D and 3D images. They allow a denoising and smoothing of image intensities while retaining and enhancing edges. As time evolves in the corresponding process, a scale of successively coarser image details is generated. Certain features, however, remain highly resolved and sharp. On the other hand, compression is an important topic in image processing as well. Here a method is presented which combines the two aspects in an efficient way. It is based on a semi–implicit Finite Element implementation of nonlinear diffusion. Error indicators guide a successive coarsening process. This leads to locally coarse grids in areas of resulting smooth image intensity, while enhanced edges are still resolved on fine grid levels. Special emphasis has been put on algorithmical aspects such as storage requirements and efficiency. Furthermore, a new nonlinear anisotropic diffusion method for vector field visualization is presented.
An Odyssey Into Local Refinement And Multilevel Preconditioning I: Optimality Of . . .
 SIAM J. NUMER. ANAL
, 2002
"... In this article, we examine the BramblePasciakXu (BPX) preconditioner in the setting of local 2D and 3D mesh refinement. While the available optimality results for the BPX preconditioner have been constructed primarily in the setting of uniformly refined meshes, a notable exception is the 2D resul ..."
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Cited by 21 (11 self)
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In this article, we examine the BramblePasciakXu (BPX) preconditioner in the setting of local 2D and 3D mesh refinement. While the available optimality results for the BPX preconditioner have been constructed primarily in the setting of uniformly refined meshes, a notable exception is the 2D result due to Dahmen and Kunoth, which established BPX optimality on meshes produced by a restricted class of local 2D redgreen refinement. The purpose of this article is to extend the original 2D DahmenKunoth result to several additional types of local 2D and 3D redgreen (conforming) and red (nonconforming) refinement procedures. The extensions are accomplished through a 3D extension of the 2D framework in the original DahmenKunoth work, by which the question of optimality is reduced to establishing that locally enriched finite element subspaces allow for the construction of a scaled basis which is formally Riesz stable. This construction in turn rests entirely on establishing a number of geometrical properties between neighboring simplices produced by the local refinement algorithms. These properties are then used to build Rieszstable scaled bases for use in the BPX optimality framework. Since the theoretical framework supports arbitrary spatial dimension d 1, we indicate clearly which geometrical properties, established here for several 2D and 3D local refinement procedures, must be reestablished to show BPX optimality for spatial dimension 4. Finally, we also present a simple alternative optimality proof of the BPX preconditioner on quasiuniform meshes in two and three spatial dimensions, through the use of Kfunctionals and H stability of L 2 projection for s 1. The proof techniques we use are quite general; in particular, the results require no smoothnes...
Optimality of multilevel preconditioners for local mesh refinement in three dimensions
 SIAM J. Numer. Anal
"... Abstract. In this article, we establish optimality of the Bramble–Pasciak–Xu (BPX) norm equivalence and optimality of the wavelet modified (or stabilized) hierarchical basis (WHB) preconditioner in the setting of local 3D mesh refinement. In the analysis of WHB methods, a critical first step is to e ..."
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Cited by 16 (8 self)
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Abstract. In this article, we establish optimality of the Bramble–Pasciak–Xu (BPX) norm equivalence and optimality of the wavelet modified (or stabilized) hierarchical basis (WHB) preconditioner in the setting of local 3D mesh refinement. In the analysis of WHB methods, a critical first step is to establish the optimality of BPX norm equivalence for the refinement procedures under consideration. While the available optimality results for the BPX norm have been constructed primarily in the setting of uniformly refined meshes, a notable exception is the local 2D redgreen result due to Dahmen and Kunoth. The purpose of this article is to extend this original 2D optimality result to the local 3D redgreen refinement procedure introduced by Bornemann, Erdmann, and Kornhuber, and then to use this result to extend the WHB optimality results from the quasiuniform setting to local 2D and 3D redgreen refinement scenarios. The BPX extension is reduced to establishing that locally enriched finite element subspaces allow for the construction of a scaled basis which is formally Riesz stable. This construction turns out to rest not only on the shape regularity of the refined elements, but also critically on a number of geometrical properties we establish between neighboring simplices produced by the Bornemann–Erdmann–Kornhuber (BEK) refinement procedure. It is possible to show that the number of degrees of freedom used for smoothing is bounded by a constant times the number of degrees of freedom introduced at that level of refinement, indicating that a practical, implementable version of the resulting BPX preconditioner for the BEK refinement setting has provably optimal (linear) computational complexity per iteration. An interesting implication of the optimality of the WHB preconditioner is the a priori H 1stability of the L2projection. The existing a posteriori approaches in the literature dictate a reconstruction of the mesh if such conditions cannot be satisfied. The theoretical framework employed supports arbitrary spatial dimension d ≥ 1 and requires no coefficient smoothness assumptions beyond those required for wellposedness in H 1.
The finite element approximation of the nonlinear poissonboltzmann equation
 SIAM Journal on Numerical Analysis
"... ABSTRACT. A widely used electrostatics model in the biomolecular modeling community, the nonlinear Poisson–Boltzmann equation, along with its finite element approximation, are analyzed in this paper. A regularized Poisson–Boltzmann equation is introduced as an auxiliary problem, making it possible t ..."
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Cited by 16 (11 self)
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ABSTRACT. A widely used electrostatics model in the biomolecular modeling community, the nonlinear Poisson–Boltzmann equation, along with its finite element approximation, are analyzed in this paper. A regularized Poisson–Boltzmann equation is introduced as an auxiliary problem, making it possible to study the original nonlinear equation with delta distribution sources. A priori error estimates for the finite element approximation are obtained for the regularized Poisson–Boltzmann equation based on certain quasiuniform grids in two and three dimensions. Adaptive finite element approximation through local refinement driven by an a posteriori error estimate is shown to converge. The Poisson–Boltzmann equation does not appear to have been previously studied in detail theoretically, and it is hoped that this paper will help provide molecular modelers with a better foundation for their analytical and computational work with the Poisson–Boltzmann equation. Note that this article apparently gives the first rigorous convergence result for a numerical discretization technique for the nonlinear Poisson– Boltzmann equation with delta distribution sources, and it also introduces the first provably convergent adaptive method for the equation. This last result is currently one of only a handful of existing convergence results of this type for nonlinear problems.
Data Structures And Concepts For Adaptive Finite Element Methods
, 1995
"... Zusammenfassung Data Structures and Concepts for Adaptive Finite Element Methods. The administration of strongly nonuniform, adaptively generated finite element meshes requires specialized techniques and data structures. A special data structure of this kind is described in this paper. It relies on ..."
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Cited by 11 (1 self)
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Zusammenfassung Data Structures and Concepts for Adaptive Finite Element Methods. The administration of strongly nonuniform, adaptively generated finite element meshes requires specialized techniques and data structures. A special data structure of this kind is described in this paper. It relies on points, edges and triangles as basic structures and is especially well suited for the realization of iterative solvers like the hierarchical basis or the multilevel nodal basis method. AMS Subject Classifications: 65N50, 65Y99, 65N30, 65N55, 65F10 Key words: Data structures, adaptive finite element methods. Datenstrukturen und Strategien fur adaptive Finite Elemente. Fur die Verwaltung von extrem nichtuniformen, adaptiv erzeugten Finite Element Gittern benotigt man spezielle Techniken und Datenstrukturen. Eine Datenstruktur dieser Art wird in diesem Artikel beschrieben. Basisstrukturen sind Punkte, Kanten und Dreiecke. Die Datenstruktur ist besonders zugeschnitten auf iterative Loser wie die hierarchische Basis oder die "multilevel nodal basis" Methode. 1.
Impact of Nonlinear Heat Transfer on Temperature Control in Regional Hyperthermia
 IEEE Trans. on Biomedical Engineering
, 1997
"... . We describe an optimization process specially designed for regional hyperthermia of deep seated tumors in order to achieve desired steadystate temperature distributions. A nonlinear threedimensional heat transfer model based on temperaturedependent blood perfusion is applied to predict the t ..."
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Cited by 10 (2 self)
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. We describe an optimization process specially designed for regional hyperthermia of deep seated tumors in order to achieve desired steadystate temperature distributions. A nonlinear threedimensional heat transfer model based on temperaturedependent blood perfusion is applied to predict the temperature. Using linearly implicit methods in time and adaptive multilevel finite elements in space, we are able to integrate efficiently the instationary nonlinear heat equation with high accuracy. Optimal heating is obtained by minimizing an integral object function which measures the distance between desired and model predicted temperatures. A sequence of minima is calculated from successively improved constantrate perfusion models employing a damped Newton method in an inner iteration. We compare temperature distributions for two individual patients calculated on coarse and fine spatial grids and present numerical results of optimizations for a Sigma 60 Applicator of the BSD 2000 Hype...
KASKADE 3.0  An ObjectOriented Adaptive Finite Element Code
 International workshop
, 1995
"... KASKADE 3.0 was developed for the solution of partial differential equations in one, two, or three space dimensions. Its objectoriented implementation concept is based on the programming language C++ . Adaptive finite element techniques are employed to provide solution procedures of optimal comp ..."
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Cited by 9 (0 self)
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KASKADE 3.0 was developed for the solution of partial differential equations in one, two, or three space dimensions. Its objectoriented implementation concept is based on the programming language C++ . Adaptive finite element techniques are employed to provide solution procedures of optimal computational complexity. This implies a posteriori error estimation, local mesh refinement and multilevel preconditioning. The program was designed both as a platform for further developments of adaptive multilevel codes and as a tool to tackle practical problems. Up to now we have implemented scalar problem types like stationary or transient heat conduction. The latter one is solved with the Rothe method, enabling adaptivity both in space and time. Some nonlinear phenomena like obstacle problems or twophase Stefan problems are incorporated as well. Extensions to vectorvalued functions and complex arithmetic are provided. We have implemented several iterative solvers for both symmet...
Downwind Numbering: A Robust Multigrid Method for ConvectionDiffusion Problems on Unstructured Grids
, 1995
"... this paper is to introduce and investigate a robust smoothing strategy for convectiondiffusion problems in two and three space dimensions without any assumption on the grid structure. The main tool to obtain such a robust smoother for these problems is an ordering strategy for the grid points which ..."
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Cited by 8 (1 self)
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this paper is to introduce and investigate a robust smoothing strategy for convectiondiffusion problems in two and three space dimensions without any assumption on the grid structure. The main tool to obtain such a robust smoother for these problems is an ordering strategy for the grid points which follows the flow direction and  combined with a GauSeidel type smoother  yields robust multigrid convergence for adaptively refined grids, provided the convection field is cyclefree.