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109
The Adaptive Multilevel Finite Element Solution of the PoissonBoltzmann Equation on Massively Parallel Computers
 J. COMPUT. CHEM
, 2000
"... Using new methods for the parallel solution of elliptic partial differential equations, the teraflops computing power of massively parallel computers can be leveraged to perform electrostatic calculations on large biological systems. This paper describes the adaptive multilevel finite element soluti ..."
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Cited by 90 (17 self)
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Using new methods for the parallel solution of elliptic partial differential equations, the teraflops computing power of massively parallel computers can be leveraged to perform electrostatic calculations on large biological systems. This paper describes the adaptive multilevel finite element solution of the PoissonBoltzmann equation for a microtubule on the NPACI IBM Blue Horizon supercomputer. The microtubule system is 40 nm in length and 24 nm in diameter, consists of roughly 600,000 atoms, and has a net charge of1800 e. PoissonBoltzmann calculations are performed for several processor configurations and the algorithm shows excellent parallel scaling.
A Multigrid Algorithm For The Mortar Finite Element Method
 SIAM J. NUMER. ANAL
"... The objective of this paper is to develop and analyse a multigrid algorithm for the system of equations arising from the mortar finite element discretization of second order elliptic boundary value problems. In order to establish the infsup condition for the saddle point formulation and to motivate ..."
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Cited by 59 (10 self)
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The objective of this paper is to develop and analyse a multigrid algorithm for the system of equations arising from the mortar finite element discretization of second order elliptic boundary value problems. In order to establish the infsup condition for the saddle point formulation and to motivate the subsequent treatment of the discretizations we revisit first briefly the theoretical concept of the mortar finite element method. Employing suitable meshdependent norms we verify the validity of the LBB condition for the resulting mixed method and prove an L 2 error estimate. This is the key for establishing a suitable approximation property for our multigrid convergence proof via a duality argument. In fact, we are able to verify optimal multigrid efficiency based on a smoother which is applied to the whole coupled system of equations. We conclude with several numerical tests of the proposed scheme which confirm the theoretical results and show the efficiency and the robustness of the method even in situations not covered by the theory.
A Generic Grid Interface for Parallel and Adaptive Scientific Computing. Part II: Implementation and Tests in DUNE
, 2007
"... In a companion paper [5] we introduced an abstract definition of a parallel and adaptive hierarchical grid for scientific computing. Based on this definition we derive an efficient interface specification as a set of C++ classes. This interface separates the applications from the grid data structure ..."
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Cited by 57 (23 self)
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In a companion paper [5] we introduced an abstract definition of a parallel and adaptive hierarchical grid for scientific computing. Based on this definition we derive an efficient interface specification as a set of C++ classes. This interface separates the applications from the grid data structures. Thus, user implementations become independent of the underlying grid implementation. Modern C++ template techniques are used to provide an interface implementation without big performance losses. The implementation is realized as part of the software environment DUNE [10]. Numerical tests demonstrate the flexibility and the efficiency of our approach. 1 1
Adaptive numerical treatment of elliptic systems on manifolds
 Advances in Computational Mathematics, 15(1):139
, 2001
"... ABSTRACT. Adaptive multilevel finite element methods are developed and analyzed for certain elliptic systems arising in geometric analysis and general relativity. This class of nonlinear elliptic systems of tensor equations on manifolds is first reviewed, and then adaptive multilevel finite element ..."
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Cited by 56 (25 self)
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ABSTRACT. Adaptive multilevel finite element methods are developed and analyzed for certain elliptic systems arising in geometric analysis and general relativity. This class of nonlinear elliptic systems of tensor equations on manifolds is first reviewed, and then adaptive multilevel finite element methods for approximating solutions to this class of problems are considered in some detail. Two a posteriori error indicators are derived, based on local residuals and on global linearized adjoint or dual problems. The design of Manifold Code (MC) is then discussed; MC is an adaptive multilevel finite element software package for 2 and 3manifolds developed over several years at Caltech and UC San Diego. It employs a posteriori error estimation, adaptive simplex subdivision, unstructured algebraic multilevel methods, global inexact Newton methods, and numerical continuation methods for the numerical solution of nonlinear covariant elliptic systems on 2 and 3manifolds. Some of the more interesting features of MC are described in detail, including some new ideas for topology and geometry representation in simplex meshes, and an unusual partition of unitybased method for exploiting parallel computers. A short example is then given which involves the Hamiltonian and momentum constraints in the Einstein equations, a representative nonlinear 4component covariant elliptic system on a Riemannian 3manifold which arises in general relativity. A number of operator properties and solvability results recently established are first summarized, making possible two quasioptimal a priori error estimates for Galerkin approximations which are then derived. These two results complete the theoretical framework for effective use of adaptive multilevel finite element methods. A sample calculation using the MC software is then presented.
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.
A finite element based level set method for twophase incompressible flows
 Comput. Vis. Sci
, 2004
"... Abstract. We present a method that has been developed for the efficient numerical simulation of twophase incompressible flows. For capturing the interface between the flows the level set technique is applied. The continuous model consists of the incompressible NavierStokes equations coupled with a ..."
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Cited by 24 (6 self)
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Abstract. We present a method that has been developed for the efficient numerical simulation of twophase incompressible flows. For capturing the interface between the flows the level set technique is applied. The continuous model consists of the incompressible NavierStokes equations coupled with an advection equation for the level set function. The effect of surface tension is modeled by a localized force term at the interface (socalled continuum surface force approach). For spatial discretization of velocity, pressure and the level set function conforming finite elements on a hierarchy of nested tetrahedral grids are used. In the finite element setting we can apply a special technique to the localized force term, which is based on a partial integration rule for the LaplaceBeltrami operator. Due to this approach the second order derivatives coming from the curvature can be eliminated. For the time discretization we apply a variant of the fractional step θscheme. The discrete saddle point problems that occur in each time step are solved using an inexact Uzawa method combined with multigrid techniques. For reparametrization of the level set function a new variant of the Fast Marching method is introduced. A special feature of the solver is that it combines the level set method with finite element discretization, LaplaceBeltrami partial integration, multilevel local refinement and multigrid solution techniques. All these components of the solver are described. Results of numerical experiments are presented. AMS subject classifications. 65M60, 65T10, 76D05, 76D45, 65N22 1. Introduction. In
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 22 (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...
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 21 (10 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 ...
Numerical Performance of Smoothers in Coupled Multigrid Methods for the Parallel Solution of the Incompressible NavierStokes Equations
 Int. J. Numer. Meth. Fluids
, 1999
"... In recent benchmark computations [1], coupled multigrid methods have been proven as efficient solvers for the incompressible NavierStokes equations. We present a numerical study of two classes of smoothers in the framework of coupled multigrid methods. The class of Vankatype smoothers is char ..."
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Cited by 16 (4 self)
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In recent benchmark computations [1], coupled multigrid methods have been proven as efficient solvers for the incompressible NavierStokes equations. We present a numerical study of two classes of smoothers in the framework of coupled multigrid methods. The class of Vankatype smoothers is characterized by the solution of small local linear systems of equations in a GaussSeidel manner in each smoothing step whereas the BrassSarazintype smoothers solve a large global saddle point problem. The behaviour of these smoothers with respect to computing times and parallel overhead is studied for twodimensional steady state and time dependent NavierStokes equations. Keywords. Incompressible NavierStokes equations, parallel coupled multigrid methods, Vankatype smoothers, BraessSarazintype smoothers. AMS(MOS) subject classifications. 65M55, 65N22, 65N55, 65Y05. 1 Introduction Within the last decades, various methods for the numerical solution of the incompressible ...
Adaptive Multigrid Methods for Signorini's Problem in Linear Elasticity
 Computing and Visualization in Science
, 2001
"... this paper we use a direct approach as introduced in [23, 26]. Our algorithm does not involve any regularization or dual formulation and should be considered as a descent method rather than an active set strategy. The basic idea is to minimize the energy on suitably selected ddimensional subspaces, ..."
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Cited by 16 (5 self)
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this paper we use a direct approach as introduced in [23, 26]. Our algorithm does not involve any regularization or dual formulation and should be considered as a descent method rather than an active set strategy. The basic idea is to minimize the energy on suitably selected ddimensional subspaces, where d  2, 3 is the dimension of the deformed body. In this way, we obtain nonlinear variants of successive subspace correction methods in the sense of Xu [34]. See e.g. [9, 30] for a similar approach to smooth nonlinear problems. Wellknown projected block GaufiSeidel relaxation is recovered by choosing the ddimensional subspaces spanned by the fine grid nodal basis functions associated with a fixed node. In order to increase convergence speed by better representation of the lowfrequency components of the error, we additionally minimize on subspaces spanned by functions with large support. The suitable selection of these coarse grid spaces is crucial for the efficiency of the resulting method. Our choice is based on sophisticated modifications of the multilevel nodal basis. Straightforward implementation of the resulting algorithm requires additional prolongations in order to check the constraints prescribed on the fine grid. As a consequence, the complexity of one iteration step is O(nl log n) for uniformly refined triangulations and might be even O(n) in the adaptive case. Optimal complexity of the multigrid Vcycle is recovered by approximating fine grid constraints on coarser grids using socalled monotone restrictions. This modification may slow down convergence, as long as the algebraic error is too large. In our numerical experiments we observed that initial iterates as provided by nested iteration are usually accurate enough to provide fast convergence through...