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Finite element analysis of singularly perturbed problems with discontinuous diffusion

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Abstract

This paper presents a non-symmetric interior penalty Galerkin \(( \text {NIPG})\) finite element scheme for one-dimensional singularly perturbed reaction-diffusion equations with discontinuous diffusion. Such problems often arise in the modelling of phase transitions. The solution of the considered class of problem is known to exhibit boundary and interior layers. Priori bounds are given for the solution and its derivatives and NIPG finite element scheme is constructed for the numerical approximation of the given problem. Error estimates for the proposed scheme are derived in the energy norm as well as the balanced norm. Error bounds have been proved to be uniform with respect to the perturbation parameter \(\varepsilon \). Numerical experiments are conducted in support of the theoretical estimates.

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Acknowledgements

The research of the first author is supported by the Council for Scientific and Industrial Research, New Delhi, India vide letter no. 09/045(1547)/2017-EMR-I. The research of the corresponding and the third author is supported by the Science and Engineering Research board, Department of Science and Technology, India via Grant No. SPG/2022/000063.

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Authors and Affiliations

  1. Department of Mathematics, University of Delhi, Delhi, 110007, India

    Ram Prasad Yadav & Pratima Rai

  2. Department of Mathematics, South Asian University, 110023, Delhi, India

    Kapil K. Sharma

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  1. Ram Prasad Yadav
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  2. Pratima Rai
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  3. Kapil K. Sharma
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Correspondence to Pratima Rai.

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Communicated by Frederic Valentin.

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Yadav, R.P., Rai, P. & Sharma, K.K. Finite element analysis of singularly perturbed problems with discontinuous diffusion. Comp. Appl. Math. 42, 257 (2023). https://doi.org/10.1007/s40314-023-02391-x

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  • DOI: https://doi.org/10.1007/s40314-023-02391-x

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