A HIERARCHICAL <i>A POSTERIORI</i> ERROR ESTIMATE FOR AN ADVECTION-DIFFUSION-REACTION PROBLEM

Araya, Rodolfo; Poza, Abner H.; Stephan, Ernst P.

doi:10.1142/s0218202505000674

Cited by 27 publications

(22 citation statements)

References 7 publications

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“…The construction of the a-posteriori estimator is given in §2. 4. In §2.5 we present a variety of one-dimensional numerical tests that support the theory.…”

Section: Introductionmentioning

confidence: 94%

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Robust a-posteriori estimator for advection-diffusion-reaction problems

Sangalli

2008

Math. Comp.

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Abstract. We propose an almost-robust residual-based a-posteriori estimator for the advection-diffusion-reaction model problem.The theory is developed in the one-dimensional setting. The numerical error is measured with respect to a norm which was introduced by the author in 2005 and somehow plays the role that the energy norm has with respect to symmetric and coercive differential operators. In particular, the mentioned norm possesses features that allow us to obtain a meaningful a-posteriori estimator, robust up to a log(P e) factor, where P e is the global Péclet number of the problem. Various numerical tests are performed in one dimension, to confirm the theoretical results and show that the proposed estimator performs better than the usual one known in literature.We also consider a possible two-dimensional extension of our result and only present a few basic numerical tests, indicating that the estimator seems to preserve the good features of the one-dimensional setting.

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“…The construction of the a-posteriori estimator is given in §2. 4. In §2.5 we present a variety of one-dimensional numerical tests that support the theory.…”

Section: Introductionmentioning

confidence: 94%

“…We do not follow this point of view in the paper, and just refer the interested reader to [3,4,10,17].…”

Section: Introductionmentioning

confidence: 99%

Robust a-posteriori estimator for advection-diffusion-reaction problems

Sangalli

2008

Math. Comp.

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“…It seems reasonable to assume that an adaptive computation using some kind of a posteriori error estimator (residual, hierarchical, etc.) [13,14] would exceedingly improve the accuracy of the solution.…”

Section: Maiprogs Modelmentioning

confidence: 99%

“…The aim is to derive efficient and reliable a posteriori error estimators for the test case in order to have a better prediction of the true error. Of course, these estimators can be used within adaptive algorithms leading to a faster convergence [13,14].…”

Section: Maiprogsmentioning

confidence: 99%

High precision modeling towards the 10^‐20 level

et al. 2013

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The requirements for accurate numerical simulation are increasing constantly. Modern high precision physics experiments now exceed the achievable numerical accuracy of standard commercial and scientific simulation tools. One example are optical resonators for which changes in the optical length are now commonly measured to 10 −15 precision [1]. The achievable measurement accuracy for resonators and cavities is directly influenced by changes in the distances between the optical components. If deformations in the range of 10 −15 occur, those effects cannot be modeled and analysed any more with standard methods based on double precision data types. New experimental approaches point out that the achievable experimental accuracies may improve down to the level of 10 −17 in the near future [2]. For the development and improvement of high precision resonators and the analysis of experimental data, new methods have to be developed which enable the needed level of simulation accuracy. Therefore we plan the development of new high precision algorithms for the simulation and modeling of thermo-mechanical effects with an achievable accuracy of 10 −20 . In this paper we analyse a test case and identify the problems on the way to this goal.

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“…Readers can find further references in [14][15][16]. In this work, we adopt the hierarchical approach first developed in [17] for the advection-diffusionreaction equation and extended in [18] to the generalized Stokes equations. See Bank-Weiser [19] and Bank-Smith [20] for related results.…”

Section: Introductionmentioning

confidence: 99%

On a hierarchical error estimator combined with a stabilized method for the Navier–Stokes equations

Araya

Poza

Valentin

2011

Numerical Methods Partial

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This work combines two complementary strategies for solving the steady incompressible Navier-Stokes model with a zeroth-order term, namely, a stabilized finite element method and a mesh-refinement approach based on an error estimator. First, equal order interpolation spaces are adopted to approximate both the velocity and the pressure while stability is recovered within the stabilization approach. Also designed to handle advection dominated flows under zeroth-order term influence, the stabilized method incorporates a new parameter with a threefold asymptotic behavior. Mesh adaptivity driven by a new hierarchical error estimator and built on the stabilized method is the second ingredient. The estimator construction process circumvents the saturation assumption by using an enhancing space strategy which is shown to be equivalent to the error. Several numerical tests validate the methodology.

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A HIERARCHICAL A POSTERIORI ERROR ESTIMATE FOR AN ADVECTION-DIFFUSION-REACTION PROBLEM

Cited by 27 publications

References 7 publications

Robust a-posteriori estimator for advection-diffusion-reaction problems

Robust a-posteriori estimator for advection-diffusion-reaction problems

High precision modeling towards the 10^‐20 level

On a hierarchical error estimator combined with a stabilized method for the Navier–Stokes equations

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A HIERARCHICAL A POSTERIORI ERROR ESTIMATE FOR AN ADVECTION-DIFFUSION-REACTION PROBLEM

Cited by 27 publications

References 7 publications

Robust a-posteriori estimator for advection-diffusion-reaction problems

Robust a-posteriori estimator for advection-diffusion-reaction problems

High precision modeling towards the 10‐20 level

On a hierarchical error estimator combined with a stabilized method for the Navier–Stokes equations

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High precision modeling towards the 10^‐20 level