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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 104 (28 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.
The cascadic multigrid method for elliptic problems
, 1996
"... The paper deals with certain adaptive multilevel methods at the confluence of nested multigrid methods and iterative methods based on the cascade principle of [10]. From the multigrid point of view, no correction cycles are needed; from the cascade principle view, a basic iteration method without ..."
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Cited by 54 (5 self)
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The paper deals with certain adaptive multilevel methods at the confluence of nested multigrid methods and iterative methods based on the cascade principle of [10]. From the multigrid point of view, no correction cycles are needed; from the cascade principle view, a basic iteration method without any preconditioner is used at successive refinement levels. For a prescribed error tolerance on the final level, more iterations must be spent on coarser grids in order to allow for less iterations on finer grids. A first candidate of such a cascadic multigrid method was the recently suggested cascadic conjugate gradient method of [9], in short CCG method, which used the CG method as basic iteration method on each level. In [18] it has been proven, that the CCG method is accurate with optimal complexity for elliptic problems in 2D and quasiuniform triangulations. The present paper simplifies that theory and extends it to more general basic iteration methods like the traditional multigrid smoothers. Moreover, an adaptive control strategy for the number of iterations on successive refinement levels for possibly highly nonuniform grids is worked out on the basis of a posteriori estimates. Numerical tests confirm the efficiency and robustness of the cascadic multigrid method.
A Posteriori Error Estimates for Elliptic Problems in Two and Three Space Dimensions
 SIAM J. NUMER. ANAL
, 1993
"... Let u 2 H be the exact solution of a given selfadjoint elliptic boundary value problem, which is approximated by some ~ u 2 S, S being a suitable finite element space. Efficient and reliable a posteriori estimates of the error jj u \Gamma ~ u jj, measuring the (local) quality of ~ u, play a cruci ..."
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Cited by 50 (8 self)
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Let u 2 H be the exact solution of a given selfadjoint elliptic boundary value problem, which is approximated by some ~ u 2 S, S being a suitable finite element space. Efficient and reliable a posteriori estimates of the error jj u \Gamma ~ u jj, measuring the (local) quality of ~ u, play a crucial role in termination criteria and in the adaptive refinement of the underlying mesh. A wellknown class of error estimates can be derived systematically by localizing the discretized defect problem using domain decomposition techniques. In the present paper, we provide a guideline for the theoretical analysis of such error estimates. We further clarify the relation to other concepts. Our analysis leads to new error estimates, which are specially suited to three space dimensions. The theoretical results are illustrated by numerical computations.
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 45 (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.
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 31 (12 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.
deal.II  A GeneralPurpose ObjectOriented Finite Element Library
, 2007
"... 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 27 (1 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 us 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.
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 26 (3 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...
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 26 (14 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...
Anisotropic adaptive simulation of transient flows using discontinuous Galerkin methods
 Int. J. Numer. Meth. Eng
"... An anisotropic adaptive analysis procedure based on a discontinuous Galerkin finite element discretization and local mesh modification of simplex elements is presented. The procedure is applied to transient 2 and 3dimensional problems governed by Euler’s equation. A smoothness indicator is used to ..."
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Cited by 25 (6 self)
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An anisotropic adaptive analysis procedure based on a discontinuous Galerkin finite element discretization and local mesh modification of simplex elements is presented. The procedure is applied to transient 2 and 3dimensional problems governed by Euler’s equation. A smoothness indicator is used to isolate jump features where an aligned mesh metric field in specified. The mesh metric field in smooth portions of the domain is controlled by a Hessian matrix constructed using a variational procedure to calculate the second derivatives. The transient examples included demonstrate the ability of the mesh modification procedures to effectively track evolving interacting features of general shape as they move through a domain. Copyright c 2000 John Wiley & Sons, Ltd. KEY WORDS: anisotropic adaptive, discontinuous Galerkin, mesh modification