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47
LockingFree Adaptive Mixed Finite Element Methods in Linear Elasticity
, 1998
"... Mixed finite element methods such as PEERS or the BDMS methods are designed to avoid locking for nearly incompressible materials in plane elasticity. In this paper, we establish a robust adaptive meshrefining algorithm that is rigorously based on a reliable and efficient a posteriori error estimate ..."
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Cited by 9 (3 self)
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Mixed finite element methods such as PEERS or the BDMS methods are designed to avoid locking for nearly incompressible materials in plane elasticity. In this paper, we establish a robust adaptive meshrefining algorithm that is rigorously based on a reliable and efficient a posteriori error estimate. Numerical evidence is provided for the independence of the constants in the a posteriori error bounds and for the efficiency of the adaptive meshrefining algorithm proposed. Key words: a posteriori error estimates, adaptive algorithm, reliability, mixed finite element method, locking 1 Introduction In this paper we investigate finite element solutions of the Lam'e system in linear elasticity and consider a plane elastic body with reference configuration \Omega ae R 2 and boundary @\Omega = \Gamma = \Gamma D [ \Gamma N , \Gamma D 6= ;, \Gamma N = \Gamman\Gamma D . Given a volume force f :\Omega ! R 2 and a traction g : \Gamma N ! R 2 , we seek (an approximation to) the displace...
Optimal and Robust A Posteriori Error Estimates IN L∞(L²) FOR THE APPROXIMATION OF ALLENCAHN EQUATIONS PAST SINGULARITIES
, 2009
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Energy Norm A Posteriori Error Estimation for Discontinuous Galerkin Approximations of Reactive Transport Problems
, 2003
"... Explicit a posteriori residual type error estimators in energy norm are derived for four primal DG schemes (i.e. OBBDG, NIPG, SIPG and IIPG) applied to transport in porous media with general kinetic reactions. A posteriori error estimators proposed here use directly all the available information f ..."
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Cited by 6 (1 self)
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Explicit a posteriori residual type error estimators in energy norm are derived for four primal DG schemes (i.e. OBBDG, NIPG, SIPG and IIPG) applied to transport in porous media with general kinetic reactions. A posteriori error estimators proposed here use directly all the available information from the numerical solution, and can be computed efficiently. Numerical examples are presented to demonstrate the efficiency and the effectiveness of these theoretical estimators.
A Posteriori Error Estimators for Mixed Finite Element Methods in Linear Elasticity
, 2002
"... Three a posteriori error estimators for PEERS and BDMS elements in linear elasticity are presented: one residual error estimator and two estimators based on the solution of auxiliary local problems with different boundary conditions. All of them are reliable and efficient with respect to the standar ..."
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Cited by 5 (1 self)
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Three a posteriori error estimators for PEERS and BDMS elements in linear elasticity are presented: one residual error estimator and two estimators based on the solution of auxiliary local problems with different boundary conditions. All of them are reliable and efficient with respect to the standard norm and furthermore robust for nearly incompressible materials.
A posteriori error estimates for finite element exterior calculus: The de rham complex. arXiv:1203.0803v3
, 2012
"... Abstract. Finite element exterior calculus (FEEC) has been developed over the past decade as a framework for constructing and analyzing stable and accurate numerical methods for partial differential equations by employing differential complexes. The recent work of Arnold, Falk and Winther [4] inclu ..."
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Abstract. Finite element exterior calculus (FEEC) has been developed over the past decade as a framework for constructing and analyzing stable and accurate numerical methods for partial differential equations by employing differential complexes. The recent work of Arnold, Falk and Winther [4] includes a welldeveloped theory of finite element methods for Hodge Laplace problems, including a priori error estimates. In this work we focus on developing a posteriori error estimates in which the computational error is bounded by some computable functional of the discrete solution and problem data. More precisely, we prove a posteriori error estimates of residual type for ArnoldFalkWinther mixed finite element methods for Hodgede Rham Laplace problems. While a number of previous works consider a posteriori error estimation for Maxwell’s equations and mixed formulations of the scalar Laplacian, the approach we take is distinguished by unified treatment of the various Hodge Laplace problems arising in the de Rham complex, consistent use of the language and analytical framework of differential forms, and the development of a posteriori error estimates for harmonic forms and the effects of their approximation on the resulting numerical method for the Hodge Laplacian. 1.
Twosided a posteriori error estimates for mixed formulations of elliptic problems
 SIAM J. Numer. Anal
"... The present work is devoted to the a posteriori error estimation for mixed approximations of linear selfadjoint elliptic problems. New guaranteed upper and lower bounds for the error measured in the natural product norm are derived, and the individual sharp upper bounds are obtained for approximati ..."
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The present work is devoted to the a posteriori error estimation for mixed approximations of linear selfadjoint elliptic problems. New guaranteed upper and lower bounds for the error measured in the natural product norm are derived, and the individual sharp upper bounds are obtained for approximation errors in each of the physical variables. All estimates are reliable and valid for any approximate solution from the class of admissible functions. The estimates contain only global constants depending solely on the domain geometry and the given operators. Moreover, it is shown that, after an appropriate scaling of the coordinates and the equation, the ratio of the upper and lower bounds for the error in the product norm never exceeds 3. The possible methods of finding the approximate mixed solution in the class of admissible functions are discussed. The estimates are computationally very cheap and can also be used for the indication of the local error distribution. As the applications, the diffusion problem as well as the problem of linear elasticity are considered.
Coupling Of NonConform Finite Elements And Boundary Elements II: A Posteriori Estimates And Adaptive MeshRefinement
, 1998
"... This paper is concerned with the coupling of nonconform finite element and boundary element methods in continuation of Part I (C. Carstensen, S.A. Funken: Coupling of nonconform finite elements and boundary elements I: a priori estimates), where we recast the interface model problem, introduced a ..."
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This paper is concerned with the coupling of nonconform finite element and boundary element methods in continuation of Part I (C. Carstensen, S.A. Funken: Coupling of nonconform finite elements and boundary elements I: a priori estimates), where we recast the interface model problem, introduced a coupling scheme and proved a priori error estimates. In this paper, we establish sharp a posteriori error estimates and so justify adaptive meshrefining algorithms for the efficient numerical treatment of transmission problems with the Laplacian in unbounded domains.
Weighted Clementtype interpolation and a posteriori analysis for FEM
 Berichtsreihe des Mathematischen Seminars, ChristianAlbrechtsUniversität zu Kiel
, 1997
"... One of the main mathematical tools in the residual based a posteriori error analysis is a weak interpolation operator due to Clement. Based on a partition of unity, we introduce a modified weak interpolation operator which enjoys a further orthogonality property. As a consequence, the volume contr ..."
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Cited by 3 (3 self)
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One of the main mathematical tools in the residual based a posteriori error analysis is a weak interpolation operator due to Clement. Based on a partition of unity, we introduce a modified weak interpolation operator which enjoys a further orthogonality property. As a consequence, the volume contribution in standard residual based a posteriori error estimates can be replaced by a smaller one which is generically of higher order, and so neglectible. We show applications to model problems for conform, nonconform, and mixed finite element methods.