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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 322 (25 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.
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 81 (14 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.
Stability Estimates of the Mortar Finite Element Method for 3Dimensional Problems
 EastWest J. Numer. Math
, 1998
"... This paper is concerned with the mortar finite element method for three spatial variables. The two main issues are the proof of the LBB condition based on appropriate choices of Lagrange multipliers and optimal efficiency of corresponding multigrid schemes for the whole coupled systems of equations. ..."
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Cited by 39 (3 self)
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This paper is concerned with the mortar finite element method for three spatial variables. The two main issues are the proof of the LBB condition based on appropriate choices of Lagrange multipliers and optimal efficiency of corresponding multigrid schemes for the whole coupled systems of equations. The implementation of the smoothing procedure also differs from that one used in the 2dimensional case. Key words: Mortar method, domain decomposition, saddle point problems, L 2  stability of mortar projections, multigrid algorithms, error estimates, efficiency of smoothing procedures. AMS subject classification: 65N55, 65N30, 65F10, 46E35. 1 Introduction The mortar method is a domain decomposition method with nonoverlapping subdomains, see e.g. [1, 2, 3, 6]. The matching of discretizations on adjacent subdomains is only enforced weakly which, in particular, facilitates employing different types of discretizations on different subdomains. Even in the case when only finite elements are ...
Multiplier Spaces For The Mortar Finite Element Method In Three Dimensions
 SIAM J. Numer. Anal
, 2000
"... . We consider the construction of multiplier spaces for use with the mortar finite element method in three spatial dimensions. Abstract conditions are given for the multiplier spaces which are su#cient to guarantee a stable and convergent mortar approximation. Three examples of multipliers satisfyin ..."
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Cited by 33 (2 self)
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. We consider the construction of multiplier spaces for use with the mortar finite element method in three spatial dimensions. Abstract conditions are given for the multiplier spaces which are su#cient to guarantee a stable and convergent mortar approximation. Three examples of multipliers satisfying these conditions are given. The first one is a dual basis example while the remaining two are based on finite volumes. Finally, the results of computational examples illustrating the theory are reported. 1. Introduction Domain decomposition methods have been widely used to design parallel algorithms for solving partial di#erential equations. The main idea of such methods as is wellknown is the following. The boundary value problem posed on a given domain is discretized by finite elements, finite di#erences, spectral or other approximation methods and as a result an algebraic problem is obtained. Preconditioners that can utilize parallel computer architectures are based on splitting the o...
A Multigrid Method for Nonconforming FEDiscretisations with Application to NonMatching Grids
, 1999
"... Nonconforming finite element discretisations require special care in the construction of the prolongation and restriction in the multigrid process. In this paper,a general scheme is proposed,which guarantees the approximation property. As an example, the technique is applied to the discretisation by ..."
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Cited by 22 (1 self)
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Nonconforming finite element discretisations require special care in the construction of the prolongation and restriction in the multigrid process. In this paper,a general scheme is proposed,which guarantees the approximation property. As an example, the technique is applied to the discretisation by nonmatching grids (mortar elements).
Inverse Inequalities on NonQuasiuniform Meshes and Application to the Mortar Element Method
 MATH. COMP
, 2001
"... We present a range of meshdependent inequalities for piecewise constant and continuous piecewise linear finite element functions u defined on locally refined shaperegular (but possibly nonquasiuniform) meshes. These inequalities involve norms of the form kh ff uk W s;p(\Omega\Gamma for positi ..."
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Cited by 22 (4 self)
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We present a range of meshdependent inequalities for piecewise constant and continuous piecewise linear finite element functions u defined on locally refined shaperegular (but possibly nonquasiuniform) meshes. These inequalities involve norms of the form kh ff uk W s;p(\Omega\Gamma for positive and negative s and ff, where h is a function which reflects the local mesh diameter in an appropriate way. The only global parameter involved is N , the total number of degrees of freedom in the finite element space, and we avoid estimates involving either the global maximum or minimum mesh diameter. Our inequalities include new variants of inverse inequalities as well as trace and extension theorems. They can be used in several areas of finite element analysis to extend results  previously known only for quasiuniform meshes  to the locally refined case. Here we describe applications to: (i) the theory of nonlinear approximation and (ii) the stability of the mortar element method for locally refined meshes.
Multigrid for the mortar finite element method
 SIAM J. Numer. Anal
, 1998
"... Abstract. A multigrid technique for uniformly preconditioning linear systems arising from a mortar finite element discretization of second order elliptic boundary value problems is described and analyzed. These problems are posed on domains partitioned into subdomains, each of which is independently ..."
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Cited by 22 (2 self)
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Abstract. A multigrid technique for uniformly preconditioning linear systems arising from a mortar finite element discretization of second order elliptic boundary value problems is described and analyzed. These problems are posed on domains partitioned into subdomains, each of which is independently triangulated in a multilevel fashion. The multilevel mortar finite element spaces based on such triangulations (which need not align across subdomain interfaces) are in general not nested. Suitable grid transfer operators and smoothers are developed which lead to a variable Vcycle preconditioner resulting in a uniformly preconditioned algebraic system. Computational results illustrating the theory are also presented. 1.
Multigrid Methods Based On The Unconstrained Product Space Arising From Mortar Finite Element Discretizations
 SIAM J. NUMER. ANAL
, 1999
"... The mortar finite element method allows the coupling of different discretizations across subregion boundaries. In the original mortar approach, the Lagrange multiplier space enforcing a weak continuity condition at the interfaces is defined as a modified finite element trace space. Here, we present ..."
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Cited by 19 (11 self)
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The mortar finite element method allows the coupling of different discretizations across subregion boundaries. In the original mortar approach, the Lagrange multiplier space enforcing a weak continuity condition at the interfaces is defined as a modified finite element trace space. Here, we present a new approach, where the Lagrange multiplier space is replaced by a dual space without loosing the optimality of the a priori bounds. Using the biorthogonality between the nodal basis functions of this Lagrange multiplier space and a finite element trace space, we derive an equivalent symmetric positive definite variational problem defined on the unconstrained product space. The introduction of this formulation is based on a local elimination process for the Lagrange multiplier. This equivalent approach is the starting point for the efficient iterative solution by a multigrid method. To obtain level independent convergence rates for the Wcycle, we have to define suitable level dependent...
The Mortar Finite Element Method for 3D Maxwell Equations: First Results
 SIAM J. Num. Anal
"... Abstract. In the framework of domain decomposition methods, we extend the main ideas of the mortar element method to the numerical solution of Maxwell’s equations (in wave form) by H(curl)conforming finite elements. The method we propose turns out to be a new nonconforming, nonoverlapping domain de ..."
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Cited by 13 (4 self)
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Abstract. In the framework of domain decomposition methods, we extend the main ideas of the mortar element method to the numerical solution of Maxwell’s equations (in wave form) by H(curl)conforming finite elements. The method we propose turns out to be a new nonconforming, nonoverlapping domain decomposition method where nonmatching grids are allowed at the interfaces between subdomains. A model problem is considered, the convergence of the discrete approximation is analyzed, and an error estimate is provided. The method is proven to be slightly suboptimal with a loss of a factor lnh  with respect to the degree of polynomials. In order to achieve this convergence result we nevertheless need extraregularity assumptions on the solution of the continuous problem.
Appending Boundary Conditions by Lagrange Multipliers: Analysis of the LBB Condition
 IGPM Report # 164, RWTH
, 1998
"... This paper is concerned with the analysis of discretization schemes for second order elliptic boundary value problems when essential boundary conditions are enforced with the aid of Lagrange multipliers. Specifically, we show how the validity of the LadysenskajaBabuskaBrezzi (LBB) condition for th ..."
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Cited by 12 (8 self)
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This paper is concerned with the analysis of discretization schemes for second order elliptic boundary value problems when essential boundary conditions are enforced with the aid of Lagrange multipliers. Specifically, we show how the validity of the LadysenskajaBabuskaBrezzi (LBB) condition for the corresponding saddle point problems depends on the various ingredients of the involved discretizations. The main result states that the LBB condition is satisfied whenever the discretization step length on the boundary, h \Gamma ¸ 2 \Gamma` , is somewhat bigger than the one on the domain, h\Omega ¸ 2 \Gammaj . This is quantified through constants stemming from the trace theorem, norm equivalences for the multiplier spaces on the boundary, and direct and inverse inequalities. In order to better understand the interplay of these constants, we then specialize the setting to wavelet discretizations. In this case the stability criteria can be stated solely in terms of spectral propertie...