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A posteriori error estimates for a dual finite element method for singularly perturbed reaction–diffusion problems

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Abstract

A posteriori error estimates are established for a two-step dual finite element method for singularly perturbed reaction–diffusion problems. The method can be considered as a modified least-squares finite element method. The least-squares functional is the basis for our residual-type a posteriori error estimators, which are shown to be reliable and efficient with respect to the error in an energy-type norm. Moreover, guaranteed upper bounds for the errors in the computed primary and dual variables are derived; these bounds are then used to drive an adaptive algorithm for our finite element method, yielding any desired accuracy. Our theory does not require the meshes generated to be shape-regular. Numerical experiments show the effectiveness of our a posteriori estimators.

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Acknowledgements

The authors are grateful to the anonymous referee for his/her careful reading and helpful suggestions to improve the presentation of this paper.

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Authors and Affiliations

  1. Department of Mathematics, Oklahoma State University, 401 Mathematical Sciences, Stillwater, OK, 74078, USA

    JaEun Ku

  2. Applied and Computational Mathematics Division, Beijing Computational Science Research Center, Beijing, 100193, China

    Martin Stynes

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  1. JaEun Ku
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Correspondence to Martin Stynes.

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Communicated by Rolf Stenberg.

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JaEun Ku’s research is supported by NSF Grant DMS-2208289. The work of Martin Stynes is supported in part by the National Natural Science Foundation of China under Grants 12171025 and NSAF-U2230402.

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Ku, J., Stynes, M. A posteriori error estimates for a dual finite element method for singularly perturbed reaction–diffusion problems. Bit Numer Math 64, 7 (2024). https://doi.org/10.1007/s10543-024-01008-x

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