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13
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...
Multilevel Preconditioners for Mixed Methods for Second Order Elliptic Problems
 with Appl
, 1994
"... . A new approach of constructing algebraic multilevel preconditioners for mixed finite element methods for second order elliptic problems with tensor coefficients on general geometry is proposed. The linear system arising from the mixed methods is first algebraically condensed to a symmetric, positi ..."
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Cited by 19 (14 self)
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. A new approach of constructing algebraic multilevel preconditioners for mixed finite element methods for second order elliptic problems with tensor coefficients on general geometry is proposed. The linear system arising from the mixed methods is first algebraically condensed to a symmetric, positive definite system for Lagrange multipliers, which corresponds to a linear system generated by standard nonconforming finite element methods. Algebraic multilevel preconditioners are then constructed for this system based on a triangulation of parallelepipeds into tetrahedral substructures. Explicit estimates of condition numbers and simple computational schemes are established for the constructed preconditioners. Finally, numerical results for the mixed finite element methods are presented to illustrate the present theory. Key words. mixed method, nonconforming method, multilevel preconditioner, condition number, second order elliptic problem AMS(MOS) subject classifications. 65N30, 65N22...
Multigrid and multilevel methods for nonconforming rotated Q1 elements
 Math. Comp
, 1998
"... Abstract. In this paper we study theoretical properties of multigrid algorithms and multilevel preconditioners for discretizations of secondorder elliptic problems using nonconforming rotated Q1 finite elements in two space dimensions. In particular, for the case of square partitions and the Laplac ..."
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Cited by 10 (4 self)
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Abstract. In this paper we study theoretical properties of multigrid algorithms and multilevel preconditioners for discretizations of secondorder elliptic problems using nonconforming rotated Q1 finite elements in two space dimensions. In particular, for the case of square partitions and the Laplacian we derive properties of the associated intergrid transfer operators which allow us to prove convergence of the Wcycle with any number of smoothing steps and closetooptimal condition number estimates for Vcycle preconditioners. This is in contrast to most of the other nonconforming finite element discretizations where only results for Wcycles with a sufficiently large number of smoothing steps and variable Vcycle multigrid preconditioners are available. Some numerical tests, including also a comparison with a preconditioner obtained by switching from the nonconforming rotated Q1 discretization to a discretization by conforming bilinear elements on the same partition, illustrate the theory. 1.
Multigrid Algorithms For Nonconforming And Mixed Methods For Symmetric And Nonsymmetric Problems
 SIAM J. Sci. Comput
, 1994
"... . In this paper we consider multigrid algorithms for nonconforming and mixed finite element methods for symmetric and nonsymmetric second order elliptic problems. We prove optimal convergence properties of the Wcycle multigrid algorithm and uniform condition number estimates for the variable Vcycl ..."
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Cited by 9 (5 self)
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. In this paper we consider multigrid algorithms for nonconforming and mixed finite element methods for symmetric and nonsymmetric second order elliptic problems. We prove optimal convergence properties of the Wcycle multigrid algorithm and uniform condition number estimates for the variable Vcycle preconditioner for the symmetric problem. For the nonsymmetric and/or indefinite problem, we show that a simple Vcycle multigrid iteration converges at a uniform rate provided that the coarsest level in the multilevel iteration is sufficiently fine (but independent on the number of multigrid levels). Various types of smoothers for the nonsymmetric and indefinite problem are discussed. Extensive numerical results for both symmetric and nonsymmetric problems are given to illustrate the present theories. Key words. mixed method, nonconforming method, finite elements, multigrid algorithm, convergence, symmetric problems, nonsymmetric and/or indefinite problems AMS(MOS) subject classificatio...
Analysis of Preconditioners for SaddlePoint Problems
"... Contents 1 Introduction 3 2 Preliminaries 4 2.1 De nitions and standard results . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Norm and eldofvalues equivalence . . . . . . . . . . . . . . . . . . . . . 7 2.3 Normequivalent preconditioners . . . . . . . . . . . . . . . . . . . . . . . . ..."
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Cited by 9 (2 self)
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Contents 1 Introduction 3 2 Preliminaries 4 2.1 De nitions and standard results . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Norm and eldofvalues equivalence . . . . . . . . . . . . . . . . . . . . . 7 2.3 Normequivalent preconditioners . . . . . . . . . . . . . . . . . . . . . . . . 8 3 Saddlepoint preconditioners 10 3.1 Some useful results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.2 Normequivalent preconditioners . . . . . . . . . . . . . . . . . . . . . . . . 13 4 Applications 17 4.1 Example: NavierStokes ow . . . . . . . . . . . . . . . . . . . . . . . . . . 17 4.2 Numerical experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 5 Conclusions 21 1 Introduction Finite element discretizations of partial dierential equations yield a sequence of linear systems of equations Ku = f ; (1.1) where K 2 R is a large, sparse matrix and n ! 1. Moreover, problem (1.1) inherits the stability conditions (the Babuska conditions [
Optimal preconditioning for RaviartThomas mixed formulation of secondorder elliptic problems
 SIAM J. Matrix Anal. Appl
, 2002
"... We evaluate two preconditioning strategies for the indefinite linear system obtained from RaviartThomas mixed finite element formulation of a secondorder elliptic problem with variable coefficients. In contrast to other approaches, our emphasis is on linear algebra; we establish the optimality of ..."
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Cited by 7 (3 self)
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We evaluate two preconditioning strategies for the indefinite linear system obtained from RaviartThomas mixed finite element formulation of a secondorder elliptic problem with variable coefficients. In contrast to other approaches, our emphasis is on linear algebra; we establish the optimality of the preconditioning using basic properties of the finite element matrices.
On a twolevel parallel MIC(0) preconditioning of CrouzeixRaviart nonconforming FEM systems
 Numerical Methods and Applications, Springer LNCS 2542, 2003
"... In this paper we analyze a twolevel preconditioner for finite element systems arising in approximations of second order elliptic boundary value problems by CrouzeixRaviart nonconforming triangular linear elements. This study is focused on the efficient implementation of the modified incomplete LU ..."
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Cited by 4 (2 self)
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In this paper we analyze a twolevel preconditioner for finite element systems arising in approximations of second order elliptic boundary value problems by CrouzeixRaviart nonconforming triangular linear elements. This study is focused on the efficient implementation of the modified incomplete LU factorization MIC(0) as a preconditioner in the PCG iterative method for the linear algebraic system. A special attention is given to the implementation of the method as a scalable parallel algorithm.
The Analysis Of Intergrid Transfer Operators And Multigrid Methods For Nonconforming Finite Elements
 EL. TRANS. NUMER. ANAL
, 1997
"... In this paper we first analyze intergrid transfer operators and their iterates for some nonconforming finite elements used for discretizations of second and fourthorder elliptic problems. Then two classes of multigrid methods using these elements are considered. The first class is the usual one, w ..."
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Cited by 3 (0 self)
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In this paper we first analyze intergrid transfer operators and their iterates for some nonconforming finite elements used for discretizations of second and fourthorder elliptic problems. Then two classes of multigrid methods using these elements are considered. The first class is the usual one, which uses discrete equations on all levels which are defined by the same discretization, while the second one is based on the Galerkin approach where quadratic forms over coarse grids are constructed from the quadratic form on the finest grid and the iterates of intergrid transfer operators, which we call the Galerkin multigrid method. The properties of these intergrid transfer operators are utilized for the analysis of the first class, while the properties of their iterates are exploited for the second one. Convergence results available for these two classes of multigrid methods are summarized here.
The Analysis of Multigrid Algorithms for Nonconforming and Mixed Methods for Second Order Elliptic Problems
 IMA Preprint 1277, Institute for Mathematics and its Applications
, 1994
"... . In this paper we consider multigrid algorithms for nonconforming and mixed finite element methods for second order elliptic problems on triangular and rectangular finite elements. We prove optimal convergence properties of the Wcycle multigrid algorithm and uniform condition number estimates for ..."
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Cited by 3 (0 self)
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. In this paper we consider multigrid algorithms for nonconforming and mixed finite element methods for second order elliptic problems on triangular and rectangular finite elements. We prove optimal convergence properties of the Wcycle multigrid algorithm and uniform condition number estimates for the variable Vcycle preconditioner. Lower order terms are treated, so our results also apply to parabolic equations. Key words. mixed method, nonconforming method, finite elements, multigrid algorithm, convergence, second order elliptic problem, parabolic problem AMS(MOS) subject classifications. 65N30, 65N22, 65F10 1. Introduction. In this paper we consider the Vcycle and Wcycle multigrid algorithms for numerical solution of the model problem (1:1) \Gammar \Delta (Aru) + cu = f in\Omega ; u = 0 on @\Omega ; using nonconforming and mixed finite element methods. Throughout this paper, \Omega ae IR n , n = 2; 3 is a simply connected bounded polygonal domain with the boundary @ \O...
Schur Complement Preconditioning for Elliptic Systems of Partial Differential Equations
, 2003
"... this paper. We only consider nite element discretizations here; in particular, we used mixed formulations of the nite element method (see [35], [6] for a general exposition). This approach leads to the following abstract mixed variational formulation Find (u; p) 2 V Q V Q such that a(u; v) + b ..."
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Cited by 2 (0 self)
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this paper. We only consider nite element discretizations here; in particular, we used mixed formulations of the nite element method (see [35], [6] for a general exposition). This approach leads to the following abstract mixed variational formulation Find (u; p) 2 V Q V Q such that a(u; v) + b 1 (v; p) = f(v) 8v 2 V; (5.1a) b 2 (u; q) + c(p; q) = g(q) 8q 2 Q; (5.1b) where a(; ) : V V ! R; b i (; ) : V Q ! R; c(; ) : QQ ! R are continuous bilinear forms, f(); g() are continuous linear forms and V; Q are two nite element spaces which satisfy the BabuskaBrezzi (BB) condition. V; Q are two Hilbert spaces natural to the problem. This condition guarantees the stability and wellposedness of our discretization (for details see Brezzi and Fortin [6])