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Error estimates of fictitious domain method with an \(H^1\) penalty approach for elliptic problems

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Abstract

This work is devoted to finding the error estimates of the fictitious domain method for elliptic problems defined on the simply connected domain. We embed the given domain into a larger rectangular domain to use the uniform mesh and extend the variational form of the original problem onto a rectangular domain with a modified \(H^1\) penalty approach. We address the convergence of the new penalized problem for both continuous and discrete cases, and find the error estimates in \(H^1\) and \(L^2\) norms with the order of 1/2 and 1, respectively. In addition, numerical experiments are performed to guarantee the theoretical outcomes, and numerically, we obtain the optimal order of convergence for the proposed method.

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Acknowledgements

The first author would like to express amiable thanks to Late Prof D R Patil for introducing him the mathematical analysis in his bachelor’s degree. The first author gratefully acknowledges the Council of Scientific & Industrial Research (CSIR), for the research fellowship, via file no. 09/992(0007)/2019-EMR-I. The authors also thank the Defence Institute of Advanced Technology, Pune, and DRDO for providing the research-friendly infrastructure and amenities. The authors are immensely grateful to the anonymous reviewers and the editor for their abundant guidance, which enriched this article.

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  1. Department of Applied Mathematics, Defence Institute of Advanced Technology, Pune, 411025, India

    Swapnil Kale & Debasish Pradhan

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  1. Swapnil Kale
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  2. Debasish Pradhan
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Communicated by Abimael Loula.

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Kale, S., Pradhan, D. Error estimates of fictitious domain method with an \(H^1\) penalty approach for elliptic problems. Comp. Appl. Math. 41, 27 (2022). https://doi.org/10.1007/s40314-021-01731-z

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  • DOI: https://doi.org/10.1007/s40314-021-01731-z

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