Results 1  10
of
21
A comparison of a posteriori error estimators for mixed finite element discretizations by raviartthomas elements
 MATH. COMP
, 1999
"... We consider mixed finite element discretizations of linear second order elliptic boundary value problems with respect to an adaptively generated hierarchy of possibly highly nonuniform simplicial triangulations. In particular, we present and analyze four different kinds of error estimators: a resid ..."
Abstract

Cited by 37 (5 self)
 Add to MetaCart
We consider mixed finite element discretizations of linear second order elliptic boundary value problems with respect to an adaptively generated hierarchy of possibly highly nonuniform simplicial triangulations. In particular, we present and analyze four different kinds of error estimators: a residual based estimator, a hierarchical one, error estimators relying on the solution of local subproblems and on a superconvergence result, respectively. Finally, we examine the relationship between the presented error estimators and compare their local components.
Multigrid method for H(DIV) in three dimensions
 ETNA
, 1997
"... . We are concerned with the design and analysis of a multigrid algorithm for H(div; ## elliptic linear variational problems. The discretization is based on H(div; conforming RaviartThomas elements. A thorough examination of the relevant bilinear form reveals that a separate treatment of vector ..."
Abstract

Cited by 33 (4 self)
 Add to MetaCart
(Show Context)
. We are concerned with the design and analysis of a multigrid algorithm for H(div; ## elliptic linear variational problems. The discretization is based on H(div; conforming RaviartThomas elements. A thorough examination of the relevant bilinear form reveals that a separate treatment of vector fields in the kernel of the divergence operator and its complement is paramount. We exploit the representation of discrete solenoidal vector fields as curls of finite element functions in socalled Nedelec spaces. It turns out that a combined nodal multilevel decomposition of both the RaviartThomas and Nedelec finite element spaces provides the foundation for a viable multigrid method. Its GauSeidel smoother involves an extra stage where solenoidal error components are tackled. By means of elaborate duality techniques we can show the asymptotic optimality in the case of uniform refinement. Numerical experiments confirm that the typical multigrid efficiency is actually achieved for model...
Error reduction and convergence for an adaptive mixed finite element method
 Mathematics of Computation
, 2005
"... Abstract. An adaptive mixed finite element method (AMFEM) is designed to guarantee an error reduction, also known as saturation property: after each refinement step, the error for the fine mesh is strictly smaller than the error for the coarse mesh up to oscillation terms. This error reduction prope ..."
Abstract

Cited by 31 (9 self)
 Add to MetaCart
(Show Context)
Abstract. An adaptive mixed finite element method (AMFEM) is designed to guarantee an error reduction, also known as saturation property: after each refinement step, the error for the fine mesh is strictly smaller than the error for the coarse mesh up to oscillation terms. This error reduction property is established here for the Raviart–Thomas finite element method with a reduction factor ρ<1 uniformly for the L 2 norm of the flux errors. Our result allows for linear convergence of a proper adaptive mixed finite element algorithm with respect to the number of refinement levels. The adaptive algorithm surprisingly does not require any particular mesh design, unlike the conforming finite element method. The new arguments are a discrete local efficiency and a quasiorthogonality estimate. The proof does not rely on duality or on regularity. 1.
A POSTERIORI ERROR ESTIMATES FOR LOWESTORDER MIXED FINITE ELEMENT DISCRETIZATIONS OF CONVECTIONDIFFUSIONREACTION EQUATIONS
, 2007
"... We establish residual a posteriori error estimates for lowestorder Raviart–Thomas mixed finite element discretizations of convectiondiffusionreaction equations on simplicial meshes in two or three space dimensions. The upwindmixed scheme is considered as well, and the emphasis is put on the pres ..."
Abstract

Cited by 28 (5 self)
 Add to MetaCart
We establish residual a posteriori error estimates for lowestorder Raviart–Thomas mixed finite element discretizations of convectiondiffusionreaction equations on simplicial meshes in two or three space dimensions. The upwindmixed scheme is considered as well, and the emphasis is put on the presence of an inhomogeneous and anisotropic diffusiondispersion tensor and on a possible convection dominance. Global upper bounds for the approximation error in the energy norm are derived, where in particular all constants are evaluated explicitly, so that the estimators are fully computable. Our estimators give local lower bounds for the error as well, and they hold from the cases where convection or reaction are not present to convection or reactiondominated problems; we prove that their local efficiency depends only on local variations in the coefficients and on the local Péclet number. Moreover, the developed general framework allows for asymptotic exactness and full robustness with respect to inhomogeneities and anisotropies. The main idea of the proof is a construction of a locally postprocessed approximate solution using the mean value and the flux in each element, known in the mixed finite element method, and a subsequent use of the abstract framework arising from the primal weak formulation of the continuous problem. Numerical experiments confirm the guaranteed upper bound and excellent efficiency and robustness of the derived estimators.
A Residual Based Error Estimator for Mortar Finite Element Discretizations
, 1997
"... Introduction We will consider the following model problem Lu := \Gammadiv (aru) + b u = f in\Omega ; u = 0 on \Gamma := @\Omega (1.1) where\Omega is a bounded, polygonal domain in IR 2 and f 2 L 2 (\Omega ). Furthermore, we assume a = (a ij ) 2 i;j=1 to be a symmetric, uniformly positiv ..."
Abstract

Cited by 23 (6 self)
 Add to MetaCart
Introduction We will consider the following model problem Lu := \Gammadiv (aru) + b u = f in\Omega ; u = 0 on \Gamma := @\Omega (1.1) where\Omega is a bounded, polygonal domain in IR 2 and f 2 L 2 (\Omega ). Furthermore, we assume a = (a ij ) 2 i;j=1 to be a symmetric, uniformly positive definite matrixvalued function with a ij 2 L 1 (\Omega ), 1 i; j 2, and 0 b 2 L 1 (\Omega ). The largest eigenvalue of a restricted to a subset D ae\Omega is denoted by ff<F9
A posteriori error estimates for the mortar mixed finite element method
 SIAM J. Numer. Anal
"... Abstract. Several a posteriori error estimators for mortar mixed finite element discretizations of elliptic equations are derived. A residualbased estimator provides optimal upper and lower bounds for the pressure error. An efficient and reliable estimator for the velocity and mortar pressure error ..."
Abstract

Cited by 18 (8 self)
 Add to MetaCart
(Show Context)
Abstract. Several a posteriori error estimators for mortar mixed finite element discretizations of elliptic equations are derived. A residualbased estimator provides optimal upper and lower bounds for the pressure error. An efficient and reliable estimator for the velocity and mortar pressure error is also derived, which is based on solving local (element) problems in a higherorder space. The interface fluxjump term that appears in the estimators can be used as an indicator for driving an adaptive process for the mortar grids only.
Convergence and optimality of adaptive mixed finite element methods
, 2009
"... The convergence and optimality of adaptive mixed finite element methods for the Poisson equation are established in this paper. The main difficulty for mixed finite element methods is the lack of minimization principle and thus the failure of orthogonality. A quasiorthogonality property is proved ..."
Abstract

Cited by 18 (5 self)
 Add to MetaCart
(Show Context)
The convergence and optimality of adaptive mixed finite element methods for the Poisson equation are established in this paper. The main difficulty for mixed finite element methods is the lack of minimization principle and thus the failure of orthogonality. A quasiorthogonality property is proved using the fact that the error is orthogonal to the divergence free subspace, while the part of the error that is not divergence free can be bounded by the data oscillation using a discrete stability result. This discrete stability result is also used to get a localized discrete upper bound which is crucial for the proof of the optimality of the adaptive approximation.
A multiresolution approach to homogenization and effective modal analysis of complex boundary value problems
 SIAM J. Appl. Math
, 2000
"... JSTOR is a notforprofit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about J ..."
Abstract

Cited by 14 (10 self)
 Add to MetaCart
(Show Context)
JSTOR is a notforprofit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. . The efficacy of modal analysis, however, critically depends on one's ability to predict the specific mode shapes and the associated eigenvalues. Let L(q, A) be a linear operator of the SturmLiouville type, with q(x) representing the system heterogeneity and A a parameter. A boundary value problem can be expressed formally by the equation Society for Industrial and Applied Mathematics
Multilevel Preconditioned Augmented Lagrangian Techniques for 2nd Order Mixed Problems
 Computing
, 1996
"... This paper aims to extend the method of Vassilevski and Wang to an adaptive setting where local refinement is admitted. An improved condition number estimate is proved. In addition we point out that the preconditioner from [27] is well suited to saddle point problems arising from augmented Lagrangia ..."
Abstract

Cited by 9 (5 self)
 Add to MetaCart
(Show Context)
This paper aims to extend the method of Vassilevski and Wang to an adaptive setting where local refinement is admitted. An improved condition number estimate is proved. In addition we point out that the preconditioner from [27] is well suited to saddle point problems arising from augmented Lagrangian methods. The paper is organized in the following way: The next section gives a brief survey of the construction and basic properties of lowest order RaviartThomas elements (cf. [8, 25]). Then we derive the augmented Lagrangian system and discuss some of its properties. We point out why preconditioning is indispensable and which variational problem has to be targeted. In section 4 we present the multilevel splitting of Vassilevski and Wang and explore it in the case of local refinement. Section 5 contains the proof that the smallest eigenvalue of the preconditioned problem is actually bounded from below independently on the number of refinement levels and the size of the augmented Lagrangian parameter. We do so by confirming the assumptions of Lion's lemma (cf. [3], Lemma 1). The following section is devoted to establishing a meshsize independent bound for the largest eigenvalue. The proof makes use of a strengthened CauchySchwarz inequality for the splitting under consideration. 4 R. HIPTMAIR et al. The details of the additive preconditioner are the subject of the following section. There also an improvement of the generic algorithm is suggested. The final section illustrates the behavior of the preconditioner when applied to several model problems. 2 Finite element spaces. We chose finite element spaces that are based upon a simplicial triangulation T h of a polygonally bounded , simply connected\Omega\Gamma Moreover we assume the elements to be shape regular, that i...