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On the finite difference method with penalty for numerical solution of PDEs over curved domains

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Abstract

This paper aims to decipher the elliptic PDEs defined over curved complex domains with a penalty approach in a finite difference scheme. Using the penalty approach, we utilize the finite difference scheme with the aid of uniform Cartesian mesh irrespective of the shape and convexity of the domain. We introduce the finite difference schemes with the \(H^1\) and \(L^2\) penalties and furnish the sharp convergence of the solution of the penalized problems to the solution of the original problem. Several iterative solvers are also equipped for both the \(H^1\) and \(L^2\) penalty schemes to compare the required number of iterations and CPU time to achieve the desired accuracy. The comprehensive numerical experiments are performed to uphold the mathematical results and to exhibit the efficiency of the proposed idea for practical scenarios.

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Acknowledgements

“The first author acknowledges the Council of Scientific & Industrial Research (CSIR-HRDG) for the research fellowship via file number 09/992(0007)/ 2019-EMR-I.”

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Authors and Affiliations

  1. School of Computer Engineering and Mathematical Sciences, Defence Institute of Advanced Technology, Pune, 411025, India

    Swapnil Kale & Debasish Pradhan

  2. School of Mathematics, Indian Institute of Science Education and Research, Thiruvananthapuram, 695551, India

    Swapnil Kale

  3. Department of Mathematics and Statistics, Chandmal Tarachand Bora College of Arts, Commerce and Science, Savitribai Phule Pune University, Shirur (Ghodnadi), Pune, 412210, India

    Swapnil Kale

  4. Department of Applied Mathematics, Andhra University, Visakhapatnam, 530003, India

    Madhusmita Tripathy

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  1. Swapnil Kale
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  2. Debasish Pradhan
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Correspondence to Swapnil Kale or Debasish Pradhan.

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Kale, S., Pradhan, D. & Tripathy, M. On the finite difference method with penalty for numerical solution of PDEs over curved domains. J. Appl. Math. Comput. 70, 893–915 (2024). https://doi.org/10.1007/s12190-024-01992-x

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  • DOI: https://doi.org/10.1007/s12190-024-01992-x

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