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48
UG  A Flexible Software Toolbox For Solving Partial Differential Equations
 COMPUTING AND VISUALIZATION IN SCIENCE
, 1997
"... Over the past two decades, some very efficient techniques for the numerical solution of partial differential equations have been developed. We are especially interested in adaptive local grid refinement on unstructured meshes, multigrid solvers and parallelization techniques. Up to now, these innova ..."
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Cited by 81 (20 self)
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Over the past two decades, some very efficient techniques for the numerical solution of partial differential equations have been developed. We are especially interested in adaptive local grid refinement on unstructured meshes, multigrid solvers and parallelization techniques. Up to now, these innovative techniques have been implemented mostly in university research codes and only very few commercial codes use them. There are two reasons for this. Firstly, the multigrid solution and adaptive refinement for many engineering applications are still a topic of active research and cannot be considered to be mature enough for routine application. Secondly, the implementation of all these techniques in a code with sufficient generality requires a lot of time and knowhow in different fields. UG (abbreviation for Unstructured Grids) has been designed to overcome these problems. It provides very general tools for the generation and manipulation of unstructured meshes in two and three space dime...
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...
Parallel Multigrid in an Adaptive PDE Solver Based on Hashing and SpaceFilling Curves
, 1997
"... this paper is organized as follows: In section 2 we discuss data structures for adaptive PDE solvers. Here, we suggest to use hash tables instead of the usually employed tree type data structures. Then, in section 3 we discuss the main features of the sequential adaptive multilevel solver. Section 4 ..."
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Cited by 39 (3 self)
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this paper is organized as follows: In section 2 we discuss data structures for adaptive PDE solvers. Here, we suggest to use hash tables instead of the usually employed tree type data structures. Then, in section 3 we discuss the main features of the sequential adaptive multilevel solver. Section 4 deals with the partitioning and distribution of adaptive grids with spacefilling curves and section 5 gives the main features of our new parallelized adaptive multilevel solver. In section 6 we present the results of numerical experiments on a parallel cluster computer with up to 64 processors. It is shown that our approach works nicely also for problems with severe singularities which result in locally refined meshes. Here, the work overhead for load balancing and data distribution remains only a small fraction of the overall work load. 2. DATA STRUCTURES FOR ADAPTIVE PDE SOLVERS 2.1. Adaptive Cycle
Adaptive sparse grid multilevel methods for elliptic PDEs based on finite differences
, 1998
"... We present a multilevel approach for the solution of partial differential equations. It is based on a multiscale basis which is constructed from a onedimensional multiscale basis by the tensor product approach. Together with the use of hash tables as data structure, this allows in a simple way for a ..."
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Cited by 30 (15 self)
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We present a multilevel approach for the solution of partial differential equations. It is based on a multiscale basis which is constructed from a onedimensional multiscale basis by the tensor product approach. Together with the use of hash tables as data structure, this allows in a simple way for adaptive refinement and is, due to the tensor product approach, well suited for higher dimensional problems. Also, the adaptive treatment of partial differential equations, the discretization (involving finite differences) and the solution (here by preconditioned BiCG) can be programmed easily. We describe the basic features of the method, discuss the discretization, the solution and the refinement procedures and report on the results of different numerical experiments.
An evaluation of parallel multigrid as a solver and a preconditioner for singular perturbed problems Part I: The standard grid sequence
, 1996
"... : In this paper we try to achieve hindependent convergence with preconditioned GMRES ([13]) and BiCGSTAB ([18]) for 2D singular perturbed equations. Three recently developed multigrid methods are adopted as a preconditioner. They are also used as solution methods in order to compare the performanc ..."
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Cited by 26 (7 self)
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: In this paper we try to achieve hindependent convergence with preconditioned GMRES ([13]) and BiCGSTAB ([18]) for 2D singular perturbed equations. Three recently developed multigrid methods are adopted as a preconditioner. They are also used as solution methods in order to compare the performance of the methods as solvers and as preconditioners. Two of the multigrid methods differ only in the transfer operators. One uses standard matrixdependent prolongation operators from [3], [5]. The second uses "upwind" prolongation operators, developed in [24]. Both employ the Galerkin coarse grid approximation and an alternating zebra line GaussSeidel smoother. The third method is based on the block LU decomposition of a matrix and on an approximate Schur complement. This multigrid variant is presented in [11]. All three multigrid algorithms are algebraic methods. The eigenvalue spectra of the three multigrid iteration matrices are analyzed for the equations solved on a 33 2 grid, in order...
A ParticlePartition Of Unity Method For The Solution Of Elliptic, Parabolic And Hyperbolic PDEs
 SIAM J. SCI. COMP
"... In this paper, we present a meshless discretization technique for instationary convectiondiffusion problems. It is based on operator splitting, the method of characteristics and a generalized partition of unity method. We focus on the discretization process and its quality. The method may be used a ..."
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Cited by 24 (5 self)
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In this paper, we present a meshless discretization technique for instationary convectiondiffusion problems. It is based on operator splitting, the method of characteristics and a generalized partition of unity method. We focus on the discretization process and its quality. The method may be used as an h or pversion. Even for general particle distributions, the convergence behavior of the different versions corresponds to that of the respective version of the finite element method on a uniform grid. We discuss the implementational aspects of the proposed method. Furthermore, we present the results of numerical examples, where we considered instationary convectiondiffusion, instationary diffusion, linear advection and elliptic problems.
Multilevel ILU Decomposition
, 1997
"... . In this paper, the multilevel ILU (MLILU) decomposition is introduced. During an incomplete Gaussian elimination process new matrix entries are generated such that a special ordering strategy yields distinct levels. On these levels, some smoothing steps are computed. The MLILU decomposition exists ..."
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Cited by 21 (2 self)
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. In this paper, the multilevel ILU (MLILU) decomposition is introduced. During an incomplete Gaussian elimination process new matrix entries are generated such that a special ordering strategy yields distinct levels. On these levels, some smoothing steps are computed. The MLILU decomposition exists and the corresponding iterative scheme converges for all symmetric and positive definite matrices. Convergence rates independent of the number of unknowns are shown numerically for several examples. Many numerical experiments including unsymmetric and anisotropic problems, problems with jumping coefficients as well as realistic problems are presented. They indicate a very robust convergence behavior of the MLILU method. Key words. Algebraic multigrid, filter condition, ILU decomposition, iterative method, partial differential equation, robustness, test vector. AMS subject classifications. 65F10, 65N55. 1 Introduction In this paper, we consider iterative algorithms u (i+1) = u (i) +M...
HashStorage Techniques for Adaptive Multilevel Solvers and Their Domain Decomposition Parallelization
 Domain decomposition methods 10. The 10th int. conf., Boulder, volume 218 of Contemp. Math
, 1998
"... this article remain attractive even for such a code. ..."
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Cited by 18 (6 self)
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this article remain attractive even for such a code.
The Coupling of Mixed and Conforming Finite Element Discretizations
, 1998
"... this paper, we introduce and analyze a special mortar finite element method. We restrict ourselves to the case of two disjoint subdomains, and use RaviartThomas finite elements in one subdomain and conforming finite elements in the other. In particular, this might be interesting for the coupling of ..."
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Cited by 18 (9 self)
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this paper, we introduce and analyze a special mortar finite element method. We restrict ourselves to the case of two disjoint subdomains, and use RaviartThomas finite elements in one subdomain and conforming finite elements in the other. In particular, this might be interesting for the coupling of different models and materials. Because of the different role of Dirichlet and Neumann boundary conditions a variational formulation without a Lagrange multiplier can be presented. It can be shown that no matching conditions for the discrete finite element spaces are necessary at the interface. Using static condensation, a coupling of conforming finite elements and enriched nonconforming CrouzeixRaviart elements satisfying Dirichlet boundary conditions at the interface is obtained. Then the Dirichlet problem is extended to a variational problem on the whole nonconforming ansatz space. In this step a piecewise constant Lagrange multiplier comes into play. By eliminating the local cubic bubble functions, it can be shown that this is equivalent to a standard mortar coupling between conforming and CrouzeixRaviart finite elements. Here the Lagrange multiplier lives on the side of the CrouzeixRaviart elements. And in contrast to the standard mortar P1/P1 coupling the discrete ansatz space for the Lagrange multiplier consists of piecewise constant functions instead of continuous piecewise linear functions. We note that the piecewise constant Lagrange multiplier represents an approximation of the Neumann boundary condition at the interface. Finally, we present some numerical results and sketch the ideas of the algorithm. The arising saddle point problems is be solved by multigrid techniques with transforming smoothers. The mortar methods have been introduced recently and a lot of ...
Simplicial Grid Refinement: On Freudenthal's Algorithm and the Optimal Number of Congruence Classes
 NUMER. MATH
, 1998
"... In the present paper we investigate Freudenthal's simplex refinement algorithm which can be considered to be the canonical generalization of Bank's well known red refinement strategy for triangles. Freudenthal's algorithm subdivides any given (n)simplex into 2 n subsimplices, in such a way that r ..."
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Cited by 17 (0 self)
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In the present paper we investigate Freudenthal's simplex refinement algorithm which can be considered to be the canonical generalization of Bank's well known red refinement strategy for triangles. Freudenthal's algorithm subdivides any given (n)simplex into 2 n subsimplices, in such a way that recursive application results in a stable hierarchy of consistent triangulations. Our investigations concentrate in particular on the number of congruence classes generated by recursive refinements. After presentation of the method and the basic ideas behind it, we will show that Freudenthal's algorithm produces at most n!=2 congruence classes for any initial (n)simplex, no matter how many subsequent refinements are performed. Moreover, we will show that this number is optimal in the sense that recursive application of any affine invariant refinement strategy with 2 n sons per element results in at least n!=2 congruence classes for almost all (n)simplices.