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19
Overlapping Schwarz Methods On Unstructured Meshes Using NonMatching Coarse Grids
 Numer. Math
, 1996
"... . We consider two level overlapping Schwarz domain decomposition methods for solving the finite element problems that arise from discretizations of elliptic problems on general unstructured meshes in two and three dimensions. Standard finite element interpolation from the coarse to the fine grid may ..."
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Cited by 49 (17 self)
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. We consider two level overlapping Schwarz domain decomposition methods for solving the finite element problems that arise from discretizations of elliptic problems on general unstructured meshes in two and three dimensions. Standard finite element interpolation from the coarse to the fine grid may be used. Our theory requires no assumption on the substructures which constitute the whole domain, so each substructure can be of arbitrary shape and of different size. The global coarse mesh is allowed to be nonnested to the fine grid on which the discrete problem is to be solved and both the coarse meshes and the fine meshes need not be quasiuniform. In addition, the domains defined by the fine and coarse grid need not be identical. The one important constraint is that the closure of the coarse grid must cover any portion of the fine grid boundary for which Neumann boundary conditions are given. In this general setting, our algorithms have the same optimal convergence rate of the usual ...
WellSpaced Points for Numerical Methods
, 1997
"... mesh generation, mesh coarsening, multigrid Abstract A numerical method for the solution of a partial differential equation (PDE) requires the following steps: (1) discretizing the domain (mesh generation); (2) using an approximation method and the mesh to transform the problem into a linear system; ..."
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Cited by 44 (2 self)
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mesh generation, mesh coarsening, multigrid Abstract A numerical method for the solution of a partial differential equation (PDE) requires the following steps: (1) discretizing the domain (mesh generation); (2) using an approximation method and the mesh to transform the problem into a linear system; (3) solving the linear system. The approximation error and convergence of the numerical method depend on the geometric quality of the mesh, which in turn depends on the size and shape of its elements. For example, the shape quality of a triangular mesh is measured by its element's aspect ratio. In this work, we shift the focus to the geometric properties of the nodes, rather than the elements, of well shaped meshes. We introduce the concept of wellspaced points and their spacing functions, and show that these enable the development of simple and efficient algorithms for the different stages of the numerical solution of PDEs. We first apply wellspaced point sets and their accompanying technology to mesh coarsening, a crucial step in the multigrid solution of a PDE. A good aspectratio coarsening sequence of an unstructured mesh M0 is a sequence of good aspectratio meshes M1; : : : ; Mk such that Mi is an approximation of Mi\Gamma 1 containing fewer nodes and elements. We present a new approach to coarsening that guarantees the sequence is also of optimal size and width up to a constant factor the first coarsening method that provides these guarantees. We also present experimental results, based on an implementation of our approach, that substantiate the theoretical claims.
Finite Element Methods and Their Convergence for Elliptic and Parabolic Interface Problems
 Numer. Math
, 1996
"... In this paper, we consider the finite element methods for solving second order elliptic and parabolic interface problems in twodimensional convex polygonal domains. Nearly the same optimal L 2 norm and energynorm error estimates as for regular problems are obtained when the interfaces are of ar ..."
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Cited by 41 (8 self)
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In this paper, we consider the finite element methods for solving second order elliptic and parabolic interface problems in twodimensional convex polygonal domains. Nearly the same optimal L 2 norm and energynorm error estimates as for regular problems are obtained when the interfaces are of arbitrary shape but are smooth, though the regularities of the solutions are low on the whole domain. The assumptions on the finite element triangulation are reasonable and practical. Mathematics Subject Classification (1991): 65N30, 65F10. A running title: Finite element methods for interface problems. Correspondence to: Dr. Jun Zou Email: zou@math.cuhk.edu.hk Fax: (852) 2603 5154 1 Institute of Mathematics, Academia Sinica, Beijing 100080, P.R. China. Email: zmchen@math03.math.ac.cn. The work of this author was partially supported by China National Natural Science Foundation. 2 Department of Mathematics, the Chinese University of Hong Kong, Shatin, N.T., Hong Kong. Email: zou@math.cuhk....
A nonoverlapping domain decomposition method for Maxwellâ€™s equations in three dimensions
 SIAM J. Numer. Anal
"... Abstract. We propose a substructuring preconditioner for solving threedimensional elliptic equations with strongly discontinuous coefficients. The new preconditioner can be viewed as a variant of the classical substructuring preconditioner proposed by Bramble, Pasiack and Schatz (1989), but with muc ..."
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Cited by 35 (10 self)
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Abstract. We propose a substructuring preconditioner for solving threedimensional elliptic equations with strongly discontinuous coefficients. The new preconditioner can be viewed as a variant of the classical substructuring preconditioner proposed by Bramble, Pasiack and Schatz (1989), but with much simpler coarse solvers. Though the condition number of the preconditioned system may not have a good bound, we are able to show that the convergence rate of the PCG method with such substructuring preconditioner is nearly optimal, and also robust with respect to the (possibly large) jumps of the coefficient in the elliptic equation. 1.
Some Nonoverlapping Domain Decomposition Methods
, 1998
"... . The purpose of this paper is to give a unified investigation of a class of nonoverlapping domain decomposition methods for solving secondorder elliptic problems in two and three dimensions. The methods under scrutiny fall into two major categories: the substructuringtype methods and the Neumann ..."
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Cited by 35 (6 self)
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. The purpose of this paper is to give a unified investigation of a class of nonoverlapping domain decomposition methods for solving secondorder elliptic problems in two and three dimensions. The methods under scrutiny fall into two major categories: the substructuringtype methods and the NeumannNeumanntype methods. The basic framework used for analysis is the parallel subspace correction method or additive Schwarz method, and other technical tools include localglobal and globallocal techniques. The analyses for both two and threedimensional cases are carried out simultaneously. Some internal relationships between various algorithms are observed and several new variants of the algorithms are also derived. Key words. nonoverlapping domain decomposition, Schur complement, localglobal and globallocal techniques, jumps in coe#cients, substructuring, NeumannNeumann, balancing methods AMS subject classifications. 65N30, 65N55, 65F10 PII. S0036144596306800 1. Introduction. T...
THE USE OF POINTWISE INTERPOLATION IN DOMAIN DECOMPOSITION METHODS WITH NONNESTED MESHES
, 1995
"... In this paper, we develop a new technique, and a corresponding theory, for Schwarz type overlapping domain decomposition methods for solving large sparse linear systems which arise from nite element discretization of elliptic partial di erential equations. The theory provides an optimal convergence ..."
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Cited by 23 (3 self)
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In this paper, we develop a new technique, and a corresponding theory, for Schwarz type overlapping domain decomposition methods for solving large sparse linear systems which arise from nite element discretization of elliptic partial di erential equations. The theory provides an optimal convergence of an additive Schwarz algorithm that is constructed with a nonnested coarse space, and a not necessarily shape regular subdomain partitioning. The theory is also applicable to the graph partitioning algorithms recently developed, [5, 15], for problems de ned on unstructured meshes.
A Ghost Cell Expansion Method for Reducing Communications in Solving PDE Problems
 PROCEEDINGS OF SC2001
, 2001
"... In solving Partial Differential Equations, such as the Barotropic equations in ocean models, on Distributed Memory Computers, finite difference methods are commonly used. Most often, processor subdomain boundaries must be updated at each time step. This boundary update process involves many messa ..."
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Cited by 15 (0 self)
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In solving Partial Differential Equations, such as the Barotropic equations in ocean models, on Distributed Memory Computers, finite difference methods are commonly used. Most often, processor subdomain boundaries must be updated at each time step. This boundary update process involves many messages of small sizes, therefore large communication overhead. Here we propose a new approach which expands the ghost cell layers and thus updates boundaries much less frequently  reducing total message volume and groupping small messages into bigger ones. Together with a technique for eliminating diagonal communications, the method speedup communication substantially, upto 170%. We explain the method and implementation in details, provide systematic timing results and performance analysis on the Cray T3E and IBM SP.
Optimal Coarsening of Unstructured Meshes
 J. Algorithms
, 1997
"... A bounded aspectratio coarsening sequence of an unstructured mesh M 0 is a sequence of meshes M 1 ; : : : ; M k such that: ffl M i is a bounded aspectratio mesh, and ffl M i is an approximation of M i\Gamma1 that has fewer elements, where a mesh is called a bounded aspectratio mesh if all it ..."
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Cited by 14 (2 self)
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A bounded aspectratio coarsening sequence of an unstructured mesh M 0 is a sequence of meshes M 1 ; : : : ; M k such that: ffl M i is a bounded aspectratio mesh, and ffl M i is an approximation of M i\Gamma1 that has fewer elements, where a mesh is called a bounded aspectratio mesh if all its elements are of bounded aspectratio. The sequence is nodenested if the set of the nodes of M i is a subset of that of M i\Gamma1 . The problem of constructing good quality coarsening sequences is a key step for hierarchical and multilevel numerical calculations. In this paper, we give an algorithm for finding a bounded aspectratio, nodenested, coarsening sequence that is of optimal size: that is, the number of meshes in the sequence, as well as the number of elements in each mesh, are within a constant factor of the smallest possible. 1 Introduction Numerical methods such as the finite element, finite difference, and finite volume methods apply the following basic steps to sol...
Numerical Identifications of Parameters in Parabolic Systems
 Inverse Problems
, 1998
"... . In this paper, we investigate the numerical identifications of physical parameters in parabolic initialboundary value problems. The identifying problem is first formulated as a constrained minimization one using the output least squares approach with the H 1 regularization or BV regularizatio ..."
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Cited by 14 (9 self)
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. In this paper, we investigate the numerical identifications of physical parameters in parabolic initialboundary value problems. The identifying problem is first formulated as a constrained minimization one using the output least squares approach with the H 1 regularization or BV regularization. Then a simple finite element method is used to approximate the constrained minimization problem and the convergence of the approximation is shown for both regularizations. The discrete constrained problem can be reduced to a sequence of unconstrained minimization problems. Numerical experiments are presented to show the efficiency of the proposed method, even for identifying highly discontinuous and oscillating parameters. 1. Introduction In this paper, we consider a finite element approach, combined with the output least squares method, for identifying the parameter q(x) in the following parabolic problem #u #t  # (q(x)#u) = f (x, t ) in # (0, T ) (1.1) with the initial condition ...
Boundary Treatments For Multilevel Methods On Unstructured Meshes
, 1996
"... . In applying multilevel iterative methods on unstructured meshes, the grid hierarchy can allow general coarse grids whose boundaries may be nonmatching to the boundary of the fine grid. In this case, the standard coarsetofine grid transfer operators with linear interpolants are not accurate enoug ..."
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Cited by 14 (8 self)
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. In applying multilevel iterative methods on unstructured meshes, the grid hierarchy can allow general coarse grids whose boundaries may be nonmatching to the boundary of the fine grid. In this case, the standard coarsetofine grid transfer operators with linear interpolants are not accurate enough at Neumann boundaries so special care is needed to correctly handle di#erent types of boundary conditions. We propose two e#ective ways to adapt the standard coarsetofine interpolations to correctly implement boundary conditions for twodimensional polygonal domains, and we provide some numerical examples of multilevel Schwarz methods (and multigrid methods) which show that these methods are as e#cient as in the structured case. In addition, we prove that the proposed interpolants possess the local optimal L 2 approximation and H 1 stability, which are essential in the convergence analysis of the multilevel Schwarz methods. Using these results, we give a condition number bound for ...