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18
A NeumannNeumann Domain Decomposition Algorithm for Solving Plate and Shell Problems
 SIAM J. NUMER. ANAL
, 1997
"... We present a new NeumannNeumann type preconditioner of large scale linear systems arising from plate and shell problems. The advantage of the new method is a smaller coarse space than those of earlier method of the authors; this improves parallel scalability. A new abstract framework for NeumannNe ..."
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Cited by 52 (8 self)
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We present a new NeumannNeumann type preconditioner of large scale linear systems arising from plate and shell problems. The advantage of the new method is a smaller coarse space than those of earlier method of the authors; this improves parallel scalability. A new abstract framework for NeumannNeumann preconditioners is used to prove almost optimal convergence properties of the method. The convergence estimates are independent of the number of subdomains, coefficient jumps between subdomains, and depend only polylogarithmically on the number of elements per subdomain. We formulate and prove an approximate parametric variational principle for ReissnerMindlin elements as the plate thickness approaches zero, which makes the results applicable to a large class of nonlocking elements in everyday engineering use. The theoretical results are confirmed by computational experiments on model problems as well as examples from real world engineering practice.
A Scalable Substructuring Method By Lagrange Multipliers For Plate Bending Problems
 SIAM J. Numer. Anal
, 1997
"... . We present a new Lagrange multiplier based domain decomposition method for solving iteratively systems of equations arising from the finite element discretization of plate bending problems. The proposed method is essentially an extension of the FETI substructuring algorithm to the biharmonic equat ..."
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Cited by 27 (13 self)
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. We present a new Lagrange multiplier based domain decomposition method for solving iteratively systems of equations arising from the finite element discretization of plate bending problems. The proposed method is essentially an extension of the FETI substructuring algorithm to the biharmonic equation. The main idea is to enforce continuity of the transversal displacement field at the subdomain crosspoints throughout the preconditioned conjugate gradient iterations. The resulting method is proved to have a condition number that does not grow with the number of subdomains, and grows at most polylogarithmically with the number of elements per subdomain. These optimal properties hold for numerous plate bending elements that are used in practice including the HCT, DKT, and a class of nonlocking elements for the ReissnerMindlin plate models. Computational experiments are reported and shown to confirm the theoretical optimal convergence properties of the new domain decomposition method. C...
Fully Discrete hpFinite Elements: Fast Quadrature
, 1999
"... A fully discrete hp nite element method is presented. It combines the features of the standard hp nite element method (conforming Galerkin Formulation, variable order quadrature schemes, geometric meshes, static condensation) and of the spectral element method (special shape functions and spect ..."
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Cited by 16 (2 self)
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A fully discrete hp nite element method is presented. It combines the features of the standard hp nite element method (conforming Galerkin Formulation, variable order quadrature schemes, geometric meshes, static condensation) and of the spectral element method (special shape functions and spectral quadrature techniques) . The speedup (relative to standard hp elements) is analyzed in detail both theoretically and computationally.
hp Discontinuous Galerkin Time Stepping For Parabolic Problems
"... The algorithmic pattern of the hp Discontinuous Galerkin Finite Element Method (DGFEM) for the time semidiscretization of abstract parabolic evolution equations is presented. In combination with a continuous hp discretization in space we obtain a fully discrete hpscheme for the numerical solution o ..."
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Cited by 11 (1 self)
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The algorithmic pattern of the hp Discontinuous Galerkin Finite Element Method (DGFEM) for the time semidiscretization of abstract parabolic evolution equations is presented. In combination with a continuous hp discretization in space we obtain a fully discrete hpscheme for the numerical solution of parabolic problems. Numerical examples for the heat equation in a two dimensional domain confirm the exponential convergence rates which are predicted by theoretical results, under realistic assumptions on the initial data and the forcing terms. We also compare different methods to reduce the computational cost of the DGFEM.
Additive Schwarz method for the hp version of the finite element method in three dimensions
 SIAM J. Numer. Anal
"... Abstract. Two additive Schwarz methods (ASMs) are proposed for the hp version of the nite element method for twodimensional elliptic problems in polygonal domains. One is based on generous overlapping of the hversion components (i.e., the linear nodal modes) and nonoverlapping of the pversion co ..."
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Cited by 8 (4 self)
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Abstract. Two additive Schwarz methods (ASMs) are proposed for the hp version of the nite element method for twodimensional elliptic problems in polygonal domains. One is based on generous overlapping of the hversion components (i.e., the linear nodal modes) and nonoverlapping of the pversion components (i.e., the highorder side modes and internal modes). Another is based on nonoverlapping for both the hversion and the pversion components. Their implementations are in parallel on the subdomain level for the hversion components and on the element level for the pversion components. The condition number for the rst method is of order O(1 + ln p)2, and for the second one is maxi(1 + ln(Hipi/hi))2, where Hi is the diameter of the subdomain Ωi, hi is the characteristic diameter of the elements in Ωi, pi is the maximum polynomial degree used in Ωi, and p = maxi pi. Key words. additive Schwarz method, the hp version, condition number, iterative and parallel solver AMS subject classications. 65F10, 65N30, 65N55 PII. S1064827595294368 1. Introduction. For
HP90: A general & flexible Fortran 90 hpFE code
, 1997
"... A general 2Dhpadaptive Finite Element #FE# implementation in Fortran 90 is described. ..."
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Cited by 4 (0 self)
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A general 2Dhpadaptive Finite Element #FE# implementation in Fortran 90 is described.
Mixed Finite Element Methods for Nonlinear Elliptic Problems: the pVersion
"... The nite element method has been used for the numerical solution of elliptic and parabolic problems for quite some time. Mixed nite element methods have now been around for nearly two decades and much literature is available for the so calledhversion of the method, based on obtaining error bounds f ..."
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Cited by 4 (0 self)
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The nite element method has been used for the numerical solution of elliptic and parabolic problems for quite some time. Mixed nite element methods have now been around for nearly two decades and much literature is available for the so calledhversion of the method, based on obtaining error bounds for approximating polynomials of a xed
Parallel Matrix Distributions: Have we been doing it all wrong?
, 1996
"... The basic premise of this report is that traditional matrix distributions for distributing matrices on distributed memory parallel architectures are in practice too restrictive. The primary problem lies with the fact that such distributions start with the matrix, not with the underlying physical pro ..."
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Cited by 3 (2 self)
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The basic premise of this report is that traditional matrix distributions for distributing matrices on distributed memory parallel architectures are in practice too restrictive. The primary problem lies with the fact that such distributions start with the matrix, not with the underlying physical problem. Through a series of examples, we show how this hampers convenient interfaces between applications and libraries. In some instances, we show how it hampers performance in general. We propose a new data distribution, Physically Based Matrix Distributions, which appear to show promise for solving the encountered problems. Some traditionally used distributions are shown to be a special, but often unnatural, case of this more general class of distributions. 1 Introduction Ever since the conception of distributed memory parallel computing, the problem of distributing data to the individual processors of a parallel computer has been of concern. Perhaps the longest studied problem has been th...
NewtonKrylov Domain Decomposition Solvers for AdaptiveApproximations of the Steady Incompressible NavierStokes Equations with Discontinuous Pressure Fields
, 53
"... this paper, we extend nonoverlapping domain decomposition techniques, previously developed for second order elliptic problems and the Stokes operator [LP96], to the solution of incompressible flow problems governed by the NavierStokes equations. In this approach, we will reduce the original proble ..."
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Cited by 1 (0 self)
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this paper, we extend nonoverlapping domain decomposition techniques, previously developed for second order elliptic problems and the Stokes operator [LP96], to the solution of incompressible flow problems governed by the NavierStokes equations. In this approach, we will reduce the original problem to a problem set on the subspace of divergence free functions, and apply existing domain decomposition techniques to the resulting subproblem. The advantage of this approach is to greatly reduce the size of the algebraic systems that have to be solved. Adaptive hp finite elements, in which the spectral order and element size are independently varied over the whole domain, are capable of delivering solution accuracies far superior to classical h\Gamma or p\Gammaversion finite element methods, for a given discretization size. Several researchers [BS87, DORH89, ROD89] have, in fact, shown that the reduction in discretization error with respect to number of unknowns can be exponential for general classes of elliptic boundary value problems, as opposed to the asymptotic algebraic rates observed for h or pversion finite element methods. Together with multiprocessor computing, these methods thus offer the possibility of ordersofmagnitude improvement in computing efficiency over existing finite element models. The principal computational cost in any finite element solution is encountered in the solver. For the nonlinear NavierStokes equations, solved using a Newton iteration scheme, the major computational cost is in the linear solve in each iterate. If time Ninth International Conference on Domain Decomposition Methods Editor Petter E. Bjrstad, Magne S. Espedal and David E. Keyes c fl1998 DDM.org 450 PATRA stepping is used to linearize the problem, the use of implicit me...
CENTER FOR COMPUTATIONAL MATHEMATICS REPORTS ITERATIVE METHODS FOR PVERSION FINITE ELEMENTS: PRECONDITIONING THIN SOLIDS
"... Abstract. We present new preconditioning strategy for thin 3D pversion elements used to model plate and shell structures as well as the microstructure of composite materials. The strategy is incorporated in an adaptive preconditioner suitable for large realworld problems. The method is shown to p ..."
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Abstract. We present new preconditioning strategy for thin 3D pversion elements used to model plate and shell structures as well as the microstructure of composite materials. The strategy is incorporated in an adaptive preconditioner suitable for large realworld problems. The method is shown to perform well for several large real world problems, including a model of the skin on an aircraft with over 1.6 million degrees of freedom, and a highly accurate model of a 28 ply laminated fiber plate. Key words. Iterative methods, pversion finite element method, plates, shells, composites, laminates 1. Introduction. The use of hierarchical high order finite elements, known as the pversion finite element method, has numerous advantages: readily available refinement by simply adding new shape functions, simplified local aposteriori error estimation, avoidance of locking, and fast convergence to the exact solution, cf., the recent monograph by Szabo and Babuska