Results 1  10
of
70
Balancing Domain Decomposition
 Comm. Numer. Meth. Engrg
, 1993
"... The NeumannNeumann algorithm is known to be an efficient domain decomposition preconditioner with unstructured subdomains for iterative solution of finite element discretizations of difficult problems with strongly discontinuous coefficients [6]. However, this algorithm suffers from the need to sol ..."
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Cited by 131 (11 self)
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The NeumannNeumann algorithm is known to be an efficient domain decomposition preconditioner with unstructured subdomains for iterative solution of finite element discretizations of difficult problems with strongly discontinuous coefficients [6]. However, this algorithm suffers from the need to solve in each iteration an inconsistent singular problem for every subdomain, and its convergence deteriorates with increasing number of subdomains due to the lack of a coarse problem to propagate the error globally. We show that the equilibrium conditions for the singular problems on subdomains lead to a simple and natural construction of a coarse problem. The construction is purely algebraic and applies also to systems, such as those that arize in elasticity. A convergence bound independent on the number of subdomains is proved and results of computational tests are reported.
Analysis of the Inexact Uzawa Algorithm for Saddle Point Problems
 SIAM J. Numer. Anal
, 1997
"... . In this paper, we consider the socalled "inexact Uzawa" algorithm for iteratively solving block saddle point problems. Such saddle point problems arise, for example, in finite element and finite difference discretizations of Stokes equations, the equations of elasticity and mixed finite element d ..."
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Cited by 70 (2 self)
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. In this paper, we consider the socalled "inexact Uzawa" algorithm for iteratively solving block saddle point problems. Such saddle point problems arise, for example, in finite element and finite difference discretizations of Stokes equations, the equations of elasticity and mixed finite element discretization of second order problems. We consider both the linear and nonlinear variants of the inexact Uzawa algorithm. We show that the linear method always converges as long as the preconditioners defining the algorithm are properly scaled. Bounds for the rate of convergence are provided in terms of the rate of convergence for the preconditioned Uzawa algorithm and the reduction factor corresponding to the preconditioner for the upper left hand block. In the nonlinear case, the inexact Uzawa algorithm is shown to converge provided that the nonlinear process approximating the inverse of the upper left hand block is of sufficient accuracy. Bounds for the nonlinear iteration are given in t...
Feti And NeumannNeumann Iterative Substructuring Methods: Connections And New Results
 Comm. Pure Appl. Math
, 1999
"... The FETI and NeumannNeumann families of algorithms are among the best known and most severely tested domain decomposition methods for elliptic partial differential equations. They are iterative substructuring methods and have many algorithmic components in common but there are also differences. The ..."
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Cited by 60 (17 self)
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The FETI and NeumannNeumann families of algorithms are among the best known and most severely tested domain decomposition methods for elliptic partial differential equations. They are iterative substructuring methods and have many algorithmic components in common but there are also differences. The purpose of this paper is to further unify the theory for these two families of methods and to introduce a new family of FETI algorithms. Bounds on the rate of convergence, which are uniform with respect to the coefficients of a family of elliptic problems with heterogeneous coefficients, are established for these new algorithms. The theory for a variant of the NeumannNeumann algorithm is also redeveloped stressing similarities to that for the FETI methods.
Balancing domain decomposition for problems with large jumps in coefficients
 Math. Comp
, 1996
"... Abstract. The Balancing Domain Decomposition algorithm uses in each iteration solution of local problems on the subdomains coupled with a coarse problem that is used to propagate the error globally and to guarantee that the possibly singular local problems are consistent. The abstract theory introdu ..."
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Cited by 57 (10 self)
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Abstract. The Balancing Domain Decomposition algorithm uses in each iteration solution of local problems on the subdomains coupled with a coarse problem that is used to propagate the error globally and to guarantee that the possibly singular local problems are consistent. The abstract theory introduced recently by the firstnamed author is used to develop condition number bounds for conforming linear elements in two and three dimensions. The bounds are independent of arbitrary coefficient jumps between subdomains and of the number of subdomains, and grow only as the squared logarithm of the mesh size h. Computational experiments for two and threedimensional problems confirm the theory. 1.
Mixed finite element methods on nonmatching multiblock grids
 SIAM J. Numer. Anal
, 2000
"... Abstract. We consider mixed finite element methods for second order elliptic equations on nonmatching multiblock grids. A mortar finite element space is introduced on the nonmatching interfaces. We approximate in this mortar space the trace of the solution, and we impose weakly a continuity of flux ..."
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Cited by 47 (24 self)
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Abstract. We consider mixed finite element methods for second order elliptic equations on nonmatching multiblock grids. A mortar finite element space is introduced on the nonmatching interfaces. We approximate in this mortar space the trace of the solution, and we impose weakly a continuity of flux condition. A standard mixed finite element method is used within the blocks. Optimal order convergence is shown for both the solution and its flux. Moreover, at certain discrete points, superconvergence is obtained for the solution and also for the flux in special cases. Computational results using an efficient parallel domain decomposition algorithm are presented in confirmation of the theory.
Enhanced CellCentered Finite Differences For Elliptic Equations On General Geometry
 SIAM J. SCI. COMPUT
, 1998
"... We present an expanded mixed finite element method for solving secondorder elliptic partial di#erential equations on geometrically general domains. For the lowestorder RaviartThomas approximating spaces, we use quadrature rules to reduce the method to cellcentered finite di#erences, possibly en ..."
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Cited by 41 (27 self)
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We present an expanded mixed finite element method for solving secondorder elliptic partial di#erential equations on geometrically general domains. For the lowestorder RaviartThomas approximating spaces, we use quadrature rules to reduce the method to cellcentered finite di#erences, possibly enhanced with some facecentered pressures. This substantially reduces the computational complexity of the problem to a symmetric, positive definite system for essentially only as many unknowns as elements. Our new method handles general shape elements (triangles, quadrilaterals, and hexahedra) and full tensor coefficients, while the standard mixed formulation reduces to finite di#erences only in special cases with rectangular elements. As in other mixed methods, we maintain the local approximation of the divergence (i.e., local mass conservation). In contrast, Galerkin finite element methods facilitate general element shapes at the cost of achieving only global mass conservation. Our method i...
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...
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 33 (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.
Preconditioning in H(div) and Applications
 Math. Comp
, 1998
"... . We consider the solution of the system of linear algebraic equations which arises from the finite element discretization of boundary value problems associated to the differential operator I \Gamma grad div. The natural setting for such problems is in the Hilbert space H(div) and the variational fo ..."
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Cited by 32 (4 self)
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. We consider the solution of the system of linear algebraic equations which arises from the finite element discretization of boundary value problems associated to the differential operator I \Gamma grad div. The natural setting for such problems is in the Hilbert space H(div) and the variational formulation is based on the inner product in H(div). We show how to construct preconditioners for these equations using both domain decomposition and multigrid techniques. These preconditioners are shown to be spectrally equivalent to the inverse of the operator. As a consequence, they may be used to precondition iterative methods so that any given error reduction may be achieved in a finite number of iterations, with the number independent of the mesh discretization. We describe applications of these results to the efficient solution of mixed and least squares finite element approximations of elliptic boundary value problems. 1. Introduction The Hilbert space H(div) consists of squareintegr...