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36
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 ..."
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Cited by 29 (4 self)
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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 NEW MULTISCALE FINITE ELEMENT METHOD FOR HIGHCONTRAST ELLIPTIC INTERFACE PROBLEMS
"... Abstract. We introduce a new multiscale finite element method which is able to accurately capture solutions of elliptic interface problems with high contrast coefficients by using only coarse quasiuniform meshes, and without resolving the interfaces. A typical application would be the modelling of f ..."
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Cited by 25 (1 self)
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Abstract. We introduce a new multiscale finite element method which is able to accurately capture solutions of elliptic interface problems with high contrast coefficients by using only coarse quasiuniform meshes, and without resolving the interfaces. A typical application would be the modelling of flow in a porous medium containing a number of inclusions of low (or high) permeability embedded in a matrix of high (respectively low) permeability. Our method is H 1 conforming, with degrees of freedom at the nodes of a triangular mesh and requires the solution of subgrid problems for the basis functions on elements which straddle the coefficient interface, but uses standard linear approximation otherwise. A key point is the introduction of novel coefficientdependent boundary conditions for the subgrid problems. Under moderate assumptions, we prove that our methods have (optimal) convergence rate of O(h) in the energy norm and O(h 2) in the L2 norm where h is the (coarse) mesh diameter and the hidden constants in these estimates are independent of the “contrast ” (i.e. ratio of largest to smallest value) of the PDE coefficient. For standard elements the best estimate in the energy norm would be O(h 1/2−ε) with a hidden constant which in general depends on the contrast. The new interior boundary conditions depend not only on the contrast of the coefficients, but also on the angles of intersection of the interface with the element edges. 1.
A unifying theory of a posteriori error control for nonconforming finite element methods
 Numer. Math
"... Abstract. Residualbased a posteriori error estimates were derived within one unifying framework for lowestorder conforming, nonconforming, and mixed finite element schemes in [C. Carstensen, Numerische Mathematik 100 (2005) 617637]. Therein, the key assumption is that the conforming firstorder f ..."
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Cited by 22 (8 self)
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Abstract. Residualbased a posteriori error estimates were derived within one unifying framework for lowestorder conforming, nonconforming, and mixed finite element schemes in [C. Carstensen, Numerische Mathematik 100 (2005) 617637]. Therein, the key assumption is that the conforming firstorder finite element space V c h annulates the linear and bounded residual ℓ written V c h ⊆ ker ℓ. That excludes particular nonconforming finite element methods (NCFEMs) on parallelograms in that V c h 6 ⊂ ker ℓ. The present paper generalises the aforementioned theory to more general situations to deduce new a posteriori error estimates, also for mortar and discontinuous Galerkin methods. The key assumption is the existence of some bounded linear operator Π: V c h → V nc h with some elementary properties. It is conjectured that the more general hypothesis (H1)(H3) can be established for all known NCFEMs. Applications on various nonstandard finite element schemes for the Laplace, Stokes, and NavierLame ́ equations illustrate the presented unifying theory of a posteriori error control for nonconforming finite element methods. 1. Unified Mixed Approach to Error Control Suppose that the primal variable u ∈ V (e.g., the displacement field) is accompanied by a dual variable p ∈ L (e.g., the flux or stress field). Typically L is some Lebesgue and V is some Sobolev space; suppose throughout this paper that L and V are Hilbert spaces and X: = L × V. Given bounded bilinear forms (1.1) a: L × L → R and b: L × V → R and well established conditions on a and b [16, 20], the linear and bounded operator
GUARANTEED AND ROBUST DISCONTINUOUS GALERKIN A POSTERIORI ERROR ESTIMATES FOR CONVECTION–DIFFUSION–REACTION PROBLEMS
 MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES
, 2008
"... We propose and study a posteriori error estimates for convection–diffusion–reaction problems with inhomogeneous and anisotropic diffusion approximated by weighted interiorpenalty discontinuous Galerkin methods. Our twofold objective is to derive estimates without undetermined constants and to anal ..."
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Cited by 20 (3 self)
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We propose and study a posteriori error estimates for convection–diffusion–reaction problems with inhomogeneous and anisotropic diffusion approximated by weighted interiorpenalty discontinuous Galerkin methods. Our twofold objective is to derive estimates without undetermined constants and to analyze carefully the robustness of the estimates in singularly perturbed regimes due to dominant convection or reaction. We first derive locally computable estimates for the error measured in the energy (semi)norm. These estimates are evaluated using H(div, Ω)conforming diffusive and convective flux reconstructions, thereby extending previous work on pure diffusion problems. The resulting estimates are semirobust in the sense that local lower error bounds can be derived using suitable cutoff functions of the local Péclet and Damköhler numbers. Fully robust estimates are obtained for the error measured in an augmented norm consisting of the energy (semi)norm, a dual norm of the skewsymmetric part of the differential operator, and a suitable contribution of the interelement jumps of the discrete solution. Numerical
Guaranteed and fully robust a posteriori error estimates . . . with discontinuous coefficients
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A POSTERIORI ENERGYNORM ERROR ESTIMATES FOR ADVECTIONDIFFUSION EQUATIONS APPROXIMATED BY WEIGHTED INTERIOR PENALTY METHODS
"... We propose and analyze a posteriori energynorm error estimates for weighted interior penalty discontinuous Galerkin approximations to advectiondiffusionreaction equations with heterogeneous and anisotropic diffusion. The weights, which play a key role in the analysis, depend on the diffusion tens ..."
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Cited by 8 (2 self)
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We propose and analyze a posteriori energynorm error estimates for weighted interior penalty discontinuous Galerkin approximations to advectiondiffusionreaction equations with heterogeneous and anisotropic diffusion. The weights, which play a key role in the analysis, depend on the diffusion tensor and are used to formulate the consistency terms in the discontinuous Galerkin method. The error upper bounds, in which all the constants are specified, consist of three terms: a residual estimator which depends only on the elementwise fluctuation of the discrete solution residual, a diffusive flux estimator where the weights used in the method enter explicitly, and a nonconforming estimator which is nonzero because of the use of discontinuous finite element spaces. The three estimators can be bounded locally by the approximation error. A particular attention is given to the dependency on problem parameters of the constants in the local lower error bounds. For moderate advection, it is shown that full robustness with respect to diffusion heterogeneities is achieved owing to the specific design of the weights in the discontinuous Galerkin method, while diffusion anisotropies remain purely local and impact the constants through the square root of the condition number of the diffusion tensor. For dominant advection, it is shown, in the spirit of previous work by Verfürth on continuous finite elements, that the constants are bounded by the square root of the local Péclet number.
Reliable a posteriori error control for nonconformal finite element approximation
 of Stokes flow, Math. Comp
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New robust nonconforming finite elements of higher order
"... We study second order nonconforming finite elements as members of a new family of higher order approaches which behave optimally not only on multilevel refined grids, but also on perturbed grids which are still shape regular but which consist no longer of asymptotically affine equivalent mesh cells. ..."
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Cited by 4 (3 self)
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We study second order nonconforming finite elements as members of a new family of higher order approaches which behave optimally not only on multilevel refined grids, but also on perturbed grids which are still shape regular but which consist no longer of asymptotically affine equivalent mesh cells. We present two approaches to prevent this order reduction: The first one is based on the use of nonparametric basis functions which are defined as polynomials on the original mesh cell. In the second approach, we define all basis functions on the reference element but add one or more nonconforming cell bubble functions which can be removed at the end by static condensation. For the last approach, we prove optimal estimates for the approximation and consistency error and derive optimal estimates for the discretization error in the case of a Poisson problem. Furthermore, we construct and analyze numerically corresponding geometrical multigrid solvers which are based on the canonical full order grid transfer operators. Based on several benchmark configurations, for scalar Poisson problems as well as for the incompressible NavierStokes equations (representing the desired application field of these nonconforming finite elements), we demonstrate the high numerical accuracy, flexibility and efficiency of the discussed new approaches which have been successfully implemented in the FeatFlow software (www.featflow.de). The presented results show that the proposed FEMmultigrid combination (together with discontinuous pressure approximations) appear to be very advantageous candidates for realistic flow simulation tools, particularly on (parallel) high performance computing systems.
Computational survey on a posteriori error estimators for nonconforming finite element methods for Poisson problems
 J. Comput. Appl. Math
, 2013
"... Abstract. This survey compares different strategies for guaranteed error control for the lowestorder nonconforming CrouzeixRaviart finite element method for the Stokes equations. The upper error bound involves the minimal distance of the computed piecewise gradient DNC uCR to the gradients of Sob ..."
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Cited by 4 (4 self)
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Abstract. This survey compares different strategies for guaranteed error control for the lowestorder nonconforming CrouzeixRaviart finite element method for the Stokes equations. The upper error bound involves the minimal distance of the computed piecewise gradient DNC uCR to the gradients of Sobolev functions with exact boundary conditions. Several improved suggestions for the cheap computation of such test functions compete in five benchmark examples. This paper provides numerical evidence that guaranteed error control of the nonconforming FEM is indeed possible for the Stokes equations with overall effectivity indices between 1 to 4. 1.