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Exponentially Accurate Spectral Element Method for Fourth Order Elliptic Problems

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Abstract

In this paper, a fully non-conforming least-squares spectral element method for fourth order elliptic problems on smooth domains is presented. The proposed method works for a general fourth order elliptic operator with non-homogeneous data in two dimensions and gives exponentially accurate solutions. We derive differentiability estimates and prove our main stability estimate theorem using a non-conforming spectral element method. We then formulate a numerical scheme using a block diagonal preconditioner. Error estimates are also proven for the proposed method. We provide the computational complexity of our method and present results of numerical simulations that have been performed to validate the theory.

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Authors and Affiliations

  1. Interdisziplinäres Zentrum für Wissenschaftliches Rechnen (IWR), Ruprecht-Karls-Universität Heidelberg, 69120, Heidelberg, Germany

    Arbaz Khan

  2. School of Engineering and Technology, BML Munjal University, Gurgaon, Haryana, 122413, India

    Akhlaq Husain

Authors
  1. Arbaz Khan
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  2. Akhlaq Husain
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Corresponding author

Correspondence to Arbaz Khan.

Additional information

The first author is thankful to the LNM Institute of Information Technology (LNMIIT), Jaipur for supporting his visits to carry out this work and also thankful to IIT Kanpur for providing the parallel computers facilities to compute the numerical results. Research is also supported by Mathematics Center Heidelberg (MATCH), Ruprecht-Karls-Universität Heidelberg, 69120 Heidelberg, Germany.

This work was carried out during the second author’s stay at the LNM Institute of Information Technology (LNMIIT), Jaipur as assistant professor.

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Khan, A., Husain, A. Exponentially Accurate Spectral Element Method for Fourth Order Elliptic Problems. J Sci Comput 71, 303–328 (2017). https://doi.org/10.1007/s10915-016-0300-z

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