Adaptive finite element methods for elliptic equations with non-smooth coefficients

Bernardi, Christine; Verfürth, Rüdiger

doi:10.1007/pl00005393

Cited by 166 publications

(192 citation statements)

References 10 publications

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“…Similar estimates have been proven in Bernardi and Verfürth [4] (see lemma 2.8 where p = 1) and in Muñoz-Sola [10] (for the Laplacian operator).…”

Section: Assumptionssupporting

confidence: 80%

“…The goal of this paper is to analyze an explicit residual-based estimator that treats the case of high order finite elements and can also handle discontinuous material coefficients. Our approach follows closely the work of Araya and Le Tallec [2] and the analysis of Bernardi and Verfürth [4], which considered source problems. The outline of the paper goes as follows.…”

Section: Introductionmentioning

confidence: 99%

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Explicit a posteriori error estimates for eigenvalue analysis of heterogeneous elastic structures

Walsh¹,

Reese²,

Hetmaniuk³

2007

Computer Methods in Applied Mechanics and Engineering

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An a posteriori error estimator is developed for the eigenvalue analysis of three-dimensional heterogeneous elastic structures. It constitutes an extension of a well-known explicit estimator to heterogeneous structures. We prove that our estimates are independent of the variations in material properties and independent of the polynomial degree of finite elements. Finally, we study numerically the effectivity of this estimator on several model problems.

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“…Similar estimates have been proven in Bernardi and Verfürth [4] (see lemma 2.8 where p = 1) and in Muñoz-Sola [10] (for the Laplacian operator).…”

Section: Assumptionssupporting

confidence: 80%

Section: Introductionmentioning

confidence: 99%

Explicit a posteriori error estimates for eigenvalue analysis of heterogeneous elastic structures

Walsh¹,

Reese²,

Hetmaniuk³

2007

Computer Methods in Applied Mechanics and Engineering

View full text Add to dashboard Cite

show abstract

“…This hypothesis implies the existence of "robust" interpolation estimates [4,8,13]. For any subset ω ⊆ Ω, let N [n] h (ω) be the set of the vertices x of the triangulation…”

Section: Definitions and General Resultsmentioning

confidence: 99%

Robusta posteriorierror estimates for finite element discretizations of the heat equation with discontinuous coefficients

Berrone¹

2006

ESAIM: M2AN

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Abstract. In this work we derive a posteriori error estimates based on equations residuals for the heat equation with discontinuous diffusivity coefficients. The estimates are based on a fully discrete scheme based on conforming finite elements in each time slab and on the A-stable θ-scheme with 1/2 ≤ θ ≤ 1. Following remarks of [Picasso, Comput. Methods Appl. Mech. Engrg. 167 (1998) 223-237; Verfürth, Calcolo 40 (2003) 195-212] it is easy to identify a time-discretization error-estimator and a spacediscretization error-estimator. In this work we introduce a similar splitting for the data-approximation error in time and in space. Assuming the quasi-monotonicity condition [Dryja et al., Numer. Math. 72 (1996) 313-348; Petzoldt, Adv. Comput. Math. 16 (2002) 47-75] we have upper and lower bounds whose ratio is independent of any meshsize, timestep, problem parameter and its jumps.Mathematics Subject Classification. 65M60, 65M15, 65M50.

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“…We present here two extensions of the previous results. First of all, following Bernardi and Verfürth [14] and Ainsworth [5] and using the averaging operator with diffusion tensor-dependent weights, one can obtain estimates robust with respect to inhomogeneities under the "monotonicity" assumption. Second, we show that our estimates are robust with respect to all inhomogeneities, anisotropies, polynomial degree, and mesh regularity for the error in the pair u h , I av (p h ) considered as an approximate solution.…”

Section: 23mentioning

confidence: 99%

Unified primal formulation-based a priori and a posteriori error analysis of mixed finite element methods

Vohralı́k

2010

Math. Comp.

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Abstract. We derive in this paper a unified framework for a priori and a posteriori error analysis of mixed finite element discretizations of second-order elliptic problems. It is based on the classical primal weak formulation, the postprocessing of the potential proposed in [T. Arbogast and Z. Chen, On the implementation of mixed methods as nonconforming methods for secondorder elliptic problems, Math. Comp. 64 (1995), 943-972], and the discrete Friedrichs inequality. Our analysis in particular avoids any explicit use of the uniform discrete inf-sup condition and in a straightforward manner and under minimal necessary assumptions, known convergence and superconvergence results are recovered. The same framework then turns out to lead to optimal a posteriori energy error bounds. In particular, estimators for all families and orders of mixed finite element methods on grids consisting of simplices or rectangular parallelepipeds are derived. They give a guaranteed and fully computable upper bound on the energy error, represent error local lower bounds, and are robust under some conditions on the diffusion-dispersion tensor. They are thus suitable for both overall error control and adaptive mesh refinement. Moreover, the developed abstract framework and a posteriori error estimates are quite general and apply to any locally conservative method. We finally prove that in parallel and simultaneously in converse to Galerkin finite element methods, under some circumstances, the weak solution is the orthogonal projection of the postprocessed mixed finite element approximation onto the H 1 0 (Ω) space and also establish several links between mixed finite element approximations and some generalized weak solutions.

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Adaptive finite element methods for elliptic equations with non-smooth coefficients

Cited by 166 publications

References 10 publications

Explicit a posteriori error estimates for eigenvalue analysis of heterogeneous elastic structures

Explicit a posteriori error estimates for eigenvalue analysis of heterogeneous elastic structures

Robusta posteriorierror estimates for finite element discretizations of the heat equation with discontinuous coefficients

Unified primal formulation-based a priori and a posteriori error analysis of mixed finite element methods

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