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118
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 54 (11 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...
An Algorithm for Coarsening Unstructured Meshes
 Numer. Math
, 1996
"... . We develop and analyze a procedure for creating a hierarchical basis of continuous piecewise linear polynomials on an arbitrary, unstructured, nonuniform triangular mesh. Using these hierarchical basis functions, we are able to define and analyze corresponding iterative methods for solving the lin ..."
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Cited by 51 (5 self)
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. We develop and analyze a procedure for creating a hierarchical basis of continuous piecewise linear polynomials on an arbitrary, unstructured, nonuniform triangular mesh. Using these hierarchical basis functions, we are able to define and analyze corresponding iterative methods for solving the linear systems arising from finite element discretizations of elliptic partial differential equations. We show that such iterative methods perform as well as those developed for the usual case of structured, locally refined meshes. In particular, we show that the generalized condition numbers for such iterative methods are of order J 2 , where J is the number of hierarchical basis levels. Key words. Finite element, hierarchical basis, multigrid, unstructured mesh. AMS subject classifications. 65F10, 65N20 1. Introduction. Iterative methods using the hierarchical basis decomposition have proved to be among the most robust for solving broad classes of elliptic partial differential equations, ...
Monotone Multigrid Methods for Elliptic Variational Inequalities I
 I. Numer. Math
, 1993
"... . We derive fast solvers for discrete elliptic variational inequalities of the first kind (obstacle problems) as resulting from the approximation of related continuous problems by piecewise linear finite elements. Using basic ideas of successive subspace correction, we modify wellknown relaxation ..."
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Cited by 50 (13 self)
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. We derive fast solvers for discrete elliptic variational inequalities of the first kind (obstacle problems) as resulting from the approximation of related continuous problems by piecewise linear finite elements. Using basic ideas of successive subspace correction, we modify wellknown relaxation methods by extending the set of search directions. Extended underrelaxations are called monotone multigrid methods, if they are quasioptimal in a certain sense. By construction, all monotone multigrid methods are globally convergent. We take a closer look at two natural variants, the standard monotone multigrid method and a truncated version. For the considered model problems, the asymptotic convergence rates resulting from the standard approach suffer from insufficient coarsegrid transport, while the truncated monotone multigrid method provides the same efficiency as in the unconstrained case. Key words: obstacle problems, adaptive finite element methods, multigrid methods AMS (MOS) subje...
A nonoverlapping domain decomposition method for Maxwell’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 48 (16 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.
Domain Decomposition Algorithms for the Partial Differential Equations of Linear Elasticity
, 1990
"... The use of the finite element method for elasticity problems results in extremely large, sparse linear systems. Historically these have been solved using direct solvers like Choleski's method. These linear systems are often illconditioned and hence require good preconditioners if they are to b ..."
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Cited by 44 (1 self)
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The use of the finite element method for elasticity problems results in extremely large, sparse linear systems. Historically these have been solved using direct solvers like Choleski's method. These linear systems are often illconditioned and hence require good preconditioners if they are to be solved iteratively. We propose and analyze three new, parallel iterative domain decomposition algorithms for the solution of these linear systems. The algorithms are also useful for other elliptic partial differential equations. Domain decomposition algorithms are designed to take advantage of a new generation of parallel computers. The domain is decomposed into overlapping or nonoverlapping subdomains. The discrete approximation to a partial differential equation is then obtained iteratively by solving problems associated with each subdomain. The algorithms are often accelerated using the conjugate gradient method. The first new algorithm presented here borrows heavily from multilevel type a...
Multilevel Algorithms Considered As Iterative Methods On Semidefinite Systems
 SIAM J. Sci. Comput
"... . For the representation of piecewise dlinear functions, instead of the usual finite element basis, we introduce a generating system that contains the nodal basis functions of the finest level and of all coarser levels of discretization. This approach enables us to work directly with multilevel dec ..."
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Cited by 43 (10 self)
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. For the representation of piecewise dlinear functions, instead of the usual finite element basis, we introduce a generating system that contains the nodal basis functions of the finest level and of all coarser levels of discretization. This approach enables us to work directly with multilevel decompositions of a function. For a partial differential equation, the Galerkin scheme based on this generating system results in a semidefinite matrix equation that has in the 1D case only about twice, in the 2D case about 4/3 times, and in the 3D case about 8/7 times, as many unknowns as the usual system. Furthermore, the semidefinite system possesses not just one, but many solutions. However, the unique solution of the usual definite finite element problem can be easily computed from every solution of the semidefinite problem. We show that modern multilevel algorithms can be considered as standard iterative methods over the semidefinite system. The conjugate gradient method (CG) for the semi...
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.
Multilevel Solvers For Unstructured Surface Meshes
 SIAM J. Sci. Comput
"... Parameterization of unstructured surface meshes is of fundamental importance in many applications of Digital Geometry Processing. Such parameterization approaches give rise to large and exceedingly illconditioned systems which are difficult or impossible to solve without the use of sophisticated mu ..."
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Cited by 38 (3 self)
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Parameterization of unstructured surface meshes is of fundamental importance in many applications of Digital Geometry Processing. Such parameterization approaches give rise to large and exceedingly illconditioned systems which are difficult or impossible to solve without the use of sophisticated multilevel preconditioning strategies. Since the underlying meshes are very fine to begin with, such multilevel preconditioners require mesh coarsening to build an appropriate hierarchy. In this paper we consider several strategies for the construction of hierarchies using ideas from mesh simplification algorithms used in the computer graphics literature. We introduce two novel hierarchy construction schemes and demonstrate their superior performance when used in conjunction with a multigrid preconditioner.
Stabilizing the Hierarchical Basis by Approximate Wavelets II: Implementation and Numerical Results
 I: Theory, Numer. Linear Alg. Appl., 4 Number
, 1998
"... This paper is the second part of a work on stabilizing the classical hierarchical basis HB by using waveletlike basis functions. Implementation techniques are of major concern for the multilevel preconditioners proposed by the authors in the first part of the work, which deals with algorithms and t ..."
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Cited by 28 (3 self)
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This paper is the second part of a work on stabilizing the classical hierarchical basis HB by using waveletlike basis functions. Implementation techniques are of major concern for the multilevel preconditioners proposed by the authors in the first part of the work, which deals with algorithms and their mathematical theory. Numerical results are presented to confirm the theory established there. A comparison of the performance of a number of multilevel methods is conducted for elliptic problems of three space variables. Key words. hierarchical basis, multilevel methods, preconditioning, finite element elliptic equations, approximate wavelets AMS subject classifications. 65F10, 65N20, 65N30 PII. S1064827596300668 1.