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A study on using hierarchical basis error estimates in anisotropic mesh adaptation for the finite element method

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Abstract

A common approach for generating an anisotropic mesh is the M-uniform mesh approach where an adaptive mesh is generated as a uniform one in the metric specified by a given tensor M. A key component is the determination of an appropriate metric, which is often based on some type of Hessian recovery. Recently, the use of a global hierarchical basis error estimator was proposed for the development of an anisotropic metric tensor for the adaptive finite element solution. This study discusses the use of this method for a selection of different applications. Numerical results show that the method performs well and is comparable with existing metric tensors based on Hessian recovery. Also, it can provide even better adaptation to the solution if applied to problems with gradient jumps and steep boundary layers. For the Poisson problem in a domain with a corner singularity, the new method provides meshes that are fully comparable to the theoretically optimal meshes.

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Notes

  1. Quadratic least squares fitting Hessian recovery [34], which seems to be the most robust and reliable Hessian recovery method [29, 32].

  2. In this paper, aspect ratio is defined as the longest edge divided by the shortest altitude. An equilateral triangle has an aspect ratio of \(2 /\sqrt{3} \approx 1.15.\)

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Acknowledgments

The author is very grateful to the anonymous referees for their valuable comments and suggestions.

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Authors and Affiliations

  1. Department of Mathematics, The University of Kansas, Lawrence, KS, USA

    Lennard Kamenski

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  1. Lennard Kamenski
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Correspondence to Lennard Kamenski.

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This research was supported in part by the German Research Foundation through the Grant KA 3215/1-1.

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Kamenski, L. A study on using hierarchical basis error estimates in anisotropic mesh adaptation for the finite element method. Engineering with Computers 28, 451–460 (2012). https://doi.org/10.1007/s00366-011-0240-z

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