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804
Unified analysis of discontinuous Galerkin methods for elliptic problems
 SIAM J. Numer. Anal
, 2001
"... Abstract. We provide a framework for the analysis of a large class of discontinuous methods for secondorder elliptic problems. It allows for the understanding and comparison of most of the discontinuous Galerkin methods that have been proposed over the past three decades for the numerical treatment ..."
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Cited by 220 (19 self)
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Abstract. We provide a framework for the analysis of a large class of discontinuous methods for secondorder elliptic problems. It allows for the understanding and comparison of most of the discontinuous Galerkin methods that have been proposed over the past three decades for the numerical treatment of elliptic problems.
Numerical solution of saddle point problems
 ACTA NUMERICA
, 2005
"... Large linear systems of saddle point type arise in a wide variety of applications throughout computational science and engineering. Due to their indefiniteness and often poor spectral properties, such linear systems represent a significant challenge for solver developers. In recent years there has b ..."
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Cited by 180 (30 self)
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Large linear systems of saddle point type arise in a wide variety of applications throughout computational science and engineering. Due to their indefiniteness and often poor spectral properties, such linear systems represent a significant challenge for solver developers. In recent years there has been a surge of interest in saddle point problems, and numerous solution techniques have been proposed for solving this type of systems. The aim of this paper is to present and discuss a large selection of solution methods for linear systems in saddle point form, with an emphasis on iterative methods for large and sparse problems.
Wavelet and Multiscale Methods for Operator Equations
 Acta Numerica
, 1997
"... this paper is to highlight some of the underlying driving analytical mechanisms. The price of a powerful tool is the effort to construct and understand it. Its successful application hinges on the realization of a number of requirements. Some space has to be reserved for a clear identification of th ..."
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Cited by 172 (40 self)
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this paper is to highlight some of the underlying driving analytical mechanisms. The price of a powerful tool is the effort to construct and understand it. Its successful application hinges on the realization of a number of requirements. Some space has to be reserved for a clear identification of these requirements as well as for their realization. This is also particularly important for understanding the severe obstructions, that keep us at present from readily materializing all the principally promising perspectives.
An overview of projection methods for incompressible flows
 Comput. Methods Appl. Mech. Engrg
"... Abstract. We introduce and study a new class of projection methods—namely, the velocitycorrection methods in standard form and in rotational form—for solving the unsteady incompressible Navier–Stokes equations. We show that the rotational form provides improved error estimates in terms of the H 1no ..."
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Cited by 90 (14 self)
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Abstract. We introduce and study a new class of projection methods—namely, the velocitycorrection methods in standard form and in rotational form—for solving the unsteady incompressible Navier–Stokes equations. We show that the rotational form provides improved error estimates in terms of the H 1norm for the velocity and of the L 2norm for the pressure. We also show that the class of fractionalstep methods introduced in [S. A. Orsag, M. Israeli, and M. Deville, J. Sci. Comput., 1 (1986), pp. 75–111] and [K. E. Karniadakis, M. Israeli, and S. A. Orsag, J. Comput. Phys., 97 (1991), pp. 414–443] can be interpreted as the rotational form of our velocitycorrection methods. Thus, to the best of our knowledge, our results provide the first rigorous proof of stability and convergence of the methods in those papers. We also emphasize that, contrary to those of the above groups, our formulations are set in the standard L 2 setting, and consequently they can be easily implemented by means of any variational approximation techniques, in particular the finite element methods. Key words. Navier–Stokes equations, projection methods, fractionalstep methods, incompressibility, finite elements, spectral approximations
Finite element exterior calculus, homological techniques, and applications
 ACTA NUMERICA
, 2006
"... ..."
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...
Fast Nonsymmetric Iterations and Preconditioning for NavierStokes Equations
 SIAM J. Sci. Comput
, 1994
"... Discretization and linearization of the steadystate NavierStokes equations gives rise to a nonsymmetric indefinite linear system of equations. In this paper, we introduce preconditioning techniques for such systems with the property that the eigenvalues of the preconditioned matrices are bounded i ..."
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Cited by 68 (9 self)
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Discretization and linearization of the steadystate NavierStokes equations gives rise to a nonsymmetric indefinite linear system of equations. In this paper, we introduce preconditioning techniques for such systems with the property that the eigenvalues of the preconditioned matrices are bounded independently of the mesh size used in the discretization. We confirm and supplement these analytic results with a series of numerical experiments indicating that Krylov subspace iterative methods for nonsymmetric systems display rates of convergence that are independent of the mesh parameter. In addition, we show that preconditioning costs can be kept small by using iterative methods for some intermediate steps performed by the preconditioner. * This work was supported by the U. S. Army Research Office under grant DAAL0392G0016 and the U. S. National Science Foundation under grant ASC8958544 at the University of Maryland, and the Science and Engineering Research Council of Great Britain V...
A leastsquares approach based on a discrete minus one inner product for first order systems
 MATH. COMP
, 1997
"... The purpose of this paper is to develop and analyze a leastsquares approximation to a first order system. The first order system represents a reformulation of a second order elliptic boundary value problem which may be indefinite and/or nonsymmetric. The approach taken here is novel in that the le ..."
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Cited by 64 (12 self)
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The purpose of this paper is to develop and analyze a leastsquares approximation to a first order system. The first order system represents a reformulation of a second order elliptic boundary value problem which may be indefinite and/or nonsymmetric. The approach taken here is novel in that the leastsquares functional employed involves a discrete inner product which is related to the inner product in H −1 (Ω) (the Sobolev space of order minus one on Ω). The use of this inner product results in a method of approximation which is optimal with respect to the required regularity as well as the order of approximation even when applied to problems with low regularity solutions. In addition, the discrete system of equations which needs to be solved in order to compute the resulting approximation is easily preconditioned, thus providing an efficient method for solving the algebraic equations. The preconditioner for this discrete system only requires the construction of preconditioners for standard second order problems, a task which is well understood.
Mixed Finite Elements for Elliptic Problems with Tensor Coefficients as CellCentered Finite Differences
 SIAM J. NUMER. ANAL
, 1997
"... We present an expanded mixed finite element approximation of second order elliptic problems containing a tensor coefficient. The mixed method is expanded in the sense that three variables are explicitly approximated, namely, the scalar unknown, the negative of its gradient, and its flux (the tensor ..."
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Cited by 63 (37 self)
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We present an expanded mixed finite element approximation of second order elliptic problems containing a tensor coefficient. The mixed method is expanded in the sense that three variables are explicitly approximated, namely, the scalar unknown, the negative of its gradient, and its flux (the tensor coefficient times the negative gradient). The resulting linear system is a saddle point problem. In the case of the lowest order RaviartThomas elements on rectangular parallelepipeds, we approximate this expanded mixed method by incorporating certain quadrature rules. This enables us to write the system as a simple, cellcentered finite difference method, requiring the solution of a sparse, positive semideflnite linear system for the scalar unknown. For a general tensor coefficient, the sparsity pattern for the scalar unknown is a nine point stencil in two dimensions, and 19 points in three dimensions. Existing theory shows that the expanded mixed method gives optimal order ap proximations in the L a and HSnorms (and superconvergence is obtained between the Laprojection of the scalar variable and its approximation). We show that these rates of convergence are retained for the finite difference method. If h denotes the maximal mesh spacing, then the optimal rate is O(h). The superconvergence rate O(h ) is obtained for the scalar unknown and rate O(h 3/) for its gradient and flux in certain discrete norms; moreover, the full O(h ) is obtained in the strict interior of the domain. Computational results illustrate these theoretical results.
A Mortar Finite Element Method Using Dual Spaces For The Lagrange Multiplier
 SIAM J. Numer. Anal
, 1998
"... The mortar finite element method allows the coupling of different discretization schemes and triangulations across subregion boundaries. In the original mortar approach the matching at the interface is realized by enforcing an orthogonality relation between the jump and a modified trace space which ..."
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Cited by 54 (8 self)
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The mortar finite element method allows the coupling of different discretization schemes and triangulations across subregion boundaries. In the original mortar approach the matching at the interface is realized by enforcing an orthogonality relation between the jump and a modified trace space which serves as a space of Lagrange multipliers. In this paper, this Lagrange multiplier space is replaced by a dual space without losing the optimality of the method. The advantage of this new approach is that the matching condition is much easier to realize. In particular, all the basis functions of the new method are supported in a few elements. The mortar map can be represented by a diagonal matrix; in the standard mortar method a linear system of equations must be solved. The problem is considered in a positive definite nonconforming variational as well as an equivalent saddlepoint formulation.