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h-p Spectral element methods for three dimensional elliptic problems on non-smooth domains, Part-I: Regularity estimates and stability theorem

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Abstract

This is the first of a series of papers devoted to the study of h-p spectral element methods for solving three dimensional elliptic boundary value problems on non-smooth domains using parallel computers. In three dimensions there are three different types of singularities namely; the vertex, the edge and the vertex-edge singularities. In addition, the solution is anisotropic in the neighbourhoods of the edges and vertex-edges. To overcome the singularities which arise in the neighbourhoods of vertices, vertex-edges and edges, we use local systems of coordinates. These local coordinates are modified versions of spherical and cylindrical coordinate systems in their respective neighbourhoods. Away from these neighbourhoods standard Cartesian coordinates are used. In each of these neighbourhoods we use a geometrical mesh which becomes finer near the corners and edges. The geometrical mesh becomes a quasi-uniform mesh in the new system of coordinates. We then derive differentiability estimates in these new set of variables and state our main stability estimate theorem using a non-conforming h-p spectral element method whose proof is given in a separate paper.

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Acknowledgements

We thank Prof. Rukmini Dey for her helpful discussions in the preparation of the manuscript. The second author is thankful to the Council of Scientific and Industrial Research (CSIR), New Delhi for supporting this research work. This work was carried out at the Department of Mathematics, Indian Institute of Technology Kanpur, Kanpur.

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Authors and Affiliations

  1. Department of Mathematics & Statistics, Indian Institute of Technology, Kanpur, 208 016, India

    P DUTT

  2. Department of Mathematics, The LNM Institute of Information Technology, Jaipur, 302 031, India

    AKHLAQ HUSAIN

  3. TIFR Centre For Applicable Mathematics, Tata Institute of Fundamental Research, Bangalore, 560 065, India

    A S VASUDEVA MURTHY

  4. Department of Aerospace Engineering, Indian Institute of Technology, Kanpur, 208 016, India

    C S UPADHYAY

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  1. P DUTT
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Correspondence to AKHLAQ HUSAIN.

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Communicating Editor: B V Rajarama Bhat

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DUTT, P., HUSAIN, A., MURTHY, A.S.V. et al. h-p Spectral element methods for three dimensional elliptic problems on non-smooth domains, Part-I: Regularity estimates and stability theorem. Proc Math Sci 125, 239–270 (2015). https://doi.org/10.1007/s12044-015-0232-9

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