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11
Adaptive Multilevel  Methods for Obstacle Problems
, 1992
"... We consider the discretization of obstacle problems for second order elliptic differential operators by piecewise linear finite elements. Assuming that the discrete problems are reduced to a sequence of linear problems by suitable active set strategies, the linear problems are solved iteratively by ..."
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Cited by 42 (6 self)
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We consider the discretization of obstacle problems for second order elliptic differential operators by piecewise linear finite elements. Assuming that the discrete problems are reduced to a sequence of linear problems by suitable active set strategies, the linear problems are solved iteratively by preconditioned cgiterations. The proposed preconditioners are treated theoretically as abstract additive Schwarz methods and are implemented as truncated hierarchical basis preconditioners. To allow for local mesh refinement we derive semilocal and local a posteriori error estimates, providing lower and upper estimates for the discretization error. The theoretical results are illustrated by numerical computations.
An overview of cache optimization techniques and cacheaware numerical algorithms
 In Proceedings of the GIDagstuhl Forschungseminar: Algorithms for Memory Hierarchies, volume 2625 of (LNCS
, 2003
"... In order to mitigate the impact of the growing gap between CPU speed and main memory performance, today's computer architectures implement hierarchical memory structures. The idea behind this approach is to hide both the low main memory bandwidth and the latency of main memory accesses which i ..."
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Cited by 38 (4 self)
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In order to mitigate the impact of the growing gap between CPU speed and main memory performance, today's computer architectures implement hierarchical memory structures. The idea behind this approach is to hide both the low main memory bandwidth and the latency of main memory accesses which is
High performance multigrid in current large scale parallel computers
 in 9th Workshop on Parallel Systems and Algorithms
, 2008
"... Abstract: Making multigrid algorithms run efficiently on large parallel computers is a challenge. Without clever data structures the communication overhead will lead to an unacceptable performance drop when using thousands of processors. We show that with a good implementation it is possible to solv ..."
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Cited by 4 (1 self)
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Abstract: Making multigrid algorithms run efficiently on large parallel computers is a challenge. Without clever data structures the communication overhead will lead to an unacceptable performance drop when using thousands of processors. We show that with a good implementation it is possible to solve a linear system with 10 11 unknowns in about 1.5 minutes on almost 10,000 processors. The data structures also allow for efficient adaptive mesh refinement, opening a wide range of applications to our solver.
Greedy and Randomized Versions of the Multiplicative Schwarz Method
, 2011
"... We consider sequential, i.e., GaussSeidel type, subspace correction methods for the iterative solution of symmetric positive definite variational problems, where the order of subspace correction steps is not deterministically fixed as in standard multiplicative Schwarz methods. Here, we greedily ch ..."
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Cited by 4 (0 self)
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We consider sequential, i.e., GaussSeidel type, subspace correction methods for the iterative solution of symmetric positive definite variational problems, where the order of subspace correction steps is not deterministically fixed as in standard multiplicative Schwarz methods. Here, we greedily choose the subspace with the largest (or at least a relatively large) residual norm for the next update step, which is also known as the GaussSouthwell method. We prove exponential convergence in the energy norm, with a reduction factor per iteration step directly related to the spectral properties, e.g., the condition number, of the underlying space splitting. To avoid the additional computational cost associated with the greedy pick, we alternatively consider choosing the next subspace randomly, and show similar estimates for the expected error reduction. We give some numerical examples, in particular applications to a Toeplitz system and to multilevel discretizations of an elliptic boundary value problem, which illustrate the theoretical estimates.
Acceleration of multigrid flow computations through dynamic adaptation of the smoothing procedure
 J. Comput. Phys
, 2000
"... The paper presents the development and investigation of an adaptivesmoothing (AS) procedure in conjunction with the full multigrid–full approximation storage method. The latter has been previously implemented by the authors [1] for the incompressible Navier–Stokes equations in conjunction with the ..."
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Cited by 2 (0 self)
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The paper presents the development and investigation of an adaptivesmoothing (AS) procedure in conjunction with the full multigrid–full approximation storage method. The latter has been previously implemented by the authors [1] for the incompressible Navier–Stokes equations in conjunction with the artificialcompressibility method and forms the basis for investigating the current AS approach. The principle of adaptive smoothing is to exploit the nonuniform convergence behavior of the numerical solution during the iterations to reduce the size of the computational domain and, subsequently, to reduce the total computing time. The implementation of the AS approach is investigated in conjunction with three different adaptivity criteria for two and threedimensional incompressible flows. Furthermore, a dynamic procedure (henceforth labeled dynamic adaptivity) for defining variably the AS parameters during the computation is also proposed and its performance is investigated in contrast to AS with constant parameters (henceforth labeled static adaptivity). Both static and dynamic adaptivity can provide similar acceleration, but the latter additionally provides more stable numerical solutions and also requires less experimentation for
Data Abstraction Techniques for Multilevel Algorithms
, 1992
"... Multilevel methods are fast and efficient solvers for a wide range of technical and scientific applications. Their structural complexity makes the construction of powerful multilevel based software difficult. Conventional software engineering concepts do not provide a sufficient basis for the implem ..."
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Cited by 1 (0 self)
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Multilevel methods are fast and efficient solvers for a wide range of technical and scientific applications. Their structural complexity makes the construction of powerful multilevel based software difficult. Conventional software engineering concepts do not provide a sufficient basis for the implementation of general, fast, and robust multilevel applications. In particular, there is a severe tradeoff between the generality of such software and its efficiency. These problems can be alleviated on the basis of a consequent data abstraction. To equally satisfy the demands for generality and efficiency it is necessary to introduce a two level software model based on a generation and an...
Nonnested and nonstructured multigrid methods applied to elastic problems. Part II: The threedimensional case
, 1998
"... this paper a review of some multigrid strategies, procedures for geometric search to implement transfer operators, expressions for calculating the number of operations and memory space, and aspects of convergence are presented. Threedimensional elastic problems are solved by multigrid, sparse Gauss ..."
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Cited by 1 (1 self)
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this paper a review of some multigrid strategies, procedures for geometric search to implement transfer operators, expressions for calculating the number of operations and memory space, and aspects of convergence are presented. Threedimensional elastic problems are solved by multigrid, sparse Gaussian elimination, and conjugate gradient methods. The number of operations and memory requirements are compared. 1 Introduction
A POSTERIORI ERROR ESTIMATES INCLUDING ALGEBRAIC ERROR: COMPUTABLE UPPER BOUNDS AND STOPPING CRITERIA FOR ITERATIVE SOLVERS
"... Abstract. We consider the finite volume and the lowestorder mixed finite element discretizations of a secondorder elliptic pure diffusion model problem. The first goal of this paper is to derive guaranteed and fully computable a posteriori error estimates which take into account an inexact solutio ..."
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Abstract. We consider the finite volume and the lowestorder mixed finite element discretizations of a secondorder elliptic pure diffusion model problem. The first goal of this paper is to derive guaranteed and fully computable a posteriori error estimates which take into account an inexact solution of the associated linear algebraic system. We show that the algebraic error can be simply bounded using the algebraic residual vector. Much better results are, however, obtained using the complementary energy of an equilibrated Raviart–Thomas–Nédélec discrete vector field whose divergence is given by a proper weighting of the residual vector. The second goal of this paper is to construct efficient stopping criteria for iterative solvers such as the conjugate gradients, GMRES, or BiCGStab. We claim that the discretization error, implied by the given numerical method, and the algebraic one should be in balance, or, more precisely, that it is enough to solve the linear algebraic system to the accuracy which guarantees that the algebraic part of the error does not contribute significantly to the whole error. Our estimates allow a reliable and cheap comparison of the discretization and algebraic errors. One can thus use them to stop the iterative algebraic solver at the desired accuracy level, without performing an excessive number of unnecessary additional iterations. Under the assumption of the relative balance between the two errors, we also prove the efficiency of our a posteriori estimates, i.e., we show that they also represent a lower bound, up to a generic constant, for the overall energy error. A local version of this result is also stated. Several numerical experiments illustrate the theoretical results. Key words. Secondorder elliptic partial differential equation, finite volume method, mixed finite element method, a posteriori error estimates, iterative methods for linear algebraic systems, stopping criteria. AMS subject classifications. 65N15, 65N30, 76M12, 65N22, 65F10
Efficient hierarchical grid coarsening for the open Poisson problem
"... The following document describes the progress made in dealing with Poisson’s equation on unbound domains. The main focus is placed on interface treatment in the tradition of Stephen McCormick’s Fast Adaptive Composite Grid method. The method itself is described in detail and numerical tests are perf ..."
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The following document describes the progress made in dealing with Poisson’s equation on unbound domains. The main focus is placed on interface treatment in the tradition of Stephen McCormick’s Fast Adaptive Composite Grid method. The method itself is described in detail and numerical tests are performed. Results of these experiments are extensively shown and analyzed in terms of convergence rates and error norms. The report shows also a way to derive the FAC from a standard approach of a totally refined grid. The focus of the report is on graphs and data for further research and publications. Contents
Uniform Convergence of . . . FOR THE TIMEHARMONIC MAXWELL EQUATION
, 2010
"... For the efficient numerical solution of indefinite linear systems arising from curl conforming edge element approximations of the timeharmonic Maxwell equation, we consider local multigrid methods (LMM) on adaptively refined meshes. The edge element discretization is done by the lowest order edge ..."
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For the efficient numerical solution of indefinite linear systems arising from curl conforming edge element approximations of the timeharmonic Maxwell equation, we consider local multigrid methods (LMM) on adaptively refined meshes. The edge element discretization is done by the lowest order edge elements of Nédélec’s first family. The LMM features local hybrid Hiptmair smoothers of Jacobi and GaussSeidel type which are performed only on basis functions associated with newly created edges/nodal points or those edges/nodal points where the support of the corresponding basis function has changed during the refinement process. The adaptive mesh refinement is based on Dörfler marking for residualtype a posteriori error estimators and the newest vertex bisection strategy. Using the abstract Schwarz theory of multilevel iterative schemes, quasioptimal convergence of the LMM is shown, i.e., the convergence rates are independent of mesh sizes and mesh levels provided the coarsest mesh is chosen sufficiently fine. The theoretical findings are illustrated by the results of some numerical examples.