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h- p Spectral element methods for three dimensional elliptic problems on non-smooth domains, Part-II: Proof of stability theorem

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Abstract

This is the second of a series of papers devoted to the study of h-p spectral element methods for three dimensional elliptic problems on non-smooth domains. The present paper addresses the proof of the main stability theorem. We assume that the differential operator is a strongly elliptic operator which satisfies Lax–Milgram conditions. The spectral element functions are non-conforming. The stability estimate theorem of this paper will be used to design a numerical scheme which give exponentially accurate solutions to three dimensional elliptic problems on non-smooth domains and can be easily implemented on parallel computers.

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References

  1. Babuska I and Guo B, Regularity of the solutions for elliptic problems on non-smooth domains in R 3, Part I: Countably normed spaces on polyhedral domains, Proc. Royal Society of Edinburgh 127A (1997) 77–126

    MathSciNet  Google Scholar 

  2. Babuska I and Guo B, Regularity of the solutions for elliptic problems on non-smooth domains in R 3, Part II: Regularity in neighbourhoods of edges, Proc. Royal Society of Edinburgh 127A (1997) 517–545

    MathSciNet  Google Scholar 

  3. Babuska I and Guo B, Approximation peroperties of the h-p version of the finite element method, Comp. Methods Appl. Mech. Engrg. 133 (1996) 319–346

    Article  MathSciNet  MATH  Google Scholar 

  4. Babuska I, Guo B, Andersson B, Oh H S and Melenk J M, Finite element methods for solving problems with singular solutions, J. Comp. Appl. Math. 74 (1996) 51–70

    Article  MathSciNet  MATH  Google Scholar 

  5. Dutt P, Akhlaq Husain, Vasudeva Murthy A S and Upadhyay C S, h-p Spectral element methods for three dimensional elliptic problems on non-smooth domains, Part-I: Regularity estimates and stability theorem, submitted for publication

  6. Dutt P, Akhlaq Husain, Vasudeva Murthy A S and Upadhyay C S, h-p Spectral element methods for three dimensional elliptic problems on non-smooth domains, Appl. Math. Comput. 234 (2014) 13–35

    Article  MathSciNet  MATH  Google Scholar 

  7. Dutt P and Tomar S, Stability estimates for h-p spectral element methods for general elliptic problems on curvilinear domains, Proc. Indian Acad. Sci. (Math. Sci.) 113 (4) (2003) 395–429

    Article  MathSciNet  MATH  Google Scholar 

  8. Grisvard P, Elliptic problems in non-smooth domains, Monographs and Studies in Mathematics 24 (1985) (Pitman Advanced Publishing Program)

  9. Guo B, h-p version of the finite element method in R 3, Theory and algorithm, in: Proceeding of ICOSAHOM’95 (eds) A Ilin and R Scott (1995) pp. 487–500

  10. Guo B, Optimal finite element approaches for elasticity problems on non-smooth domains in R 3, in: Computaional Mechanics’s (1995) (Springer) vol. 1, pp. 427–432

  11. Guo B and Oh S H, The method of auxiliary mapping for the finite element solutions of elliptic partial differential equations on non-smooth domains in R 3, available at http://home.cc.umanitoba.ca/~guo/publication.htm (1995)

  12. Guo B, The h-p version of the finite element method for solving boundary value problems in polyhedral domains, in: Boundary value problems and integral equations in non-smooth domains (eds) M Costabel, M Dauge and C Nicaise (1994) (Marcel Dekker Inc) pp. 101–120

  13. Husain Akhlaq, h-p spectral element methods for three dimensional elliptic problems on non-smooth domains using parallel computers, Ph.D. Thesis, IIT Kanpur, India, Reprint available at arXiv:1110.2316 (2011)

  14. Wolfgang Kuhnel, Differential geometry, curves-surfaces-manifolds (2005) (AMS, Student Mathematical Library) p. 96

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Acknowledgements

The authors would like to 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.

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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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  2. AKHLAQ HUSAIN
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Correspondence to AKHLAQ HUSAIN.

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DUTT, P., HUSAIN, A., VASUDEVA MURTHY, A.S. et al. h- p Spectral element methods for three dimensional elliptic problems on non-smooth domains, Part-II: Proof of stability theorem. Proc Math Sci 125, 413–447 (2015). https://doi.org/10.1007/s12044-015-0239-2

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  • DOI: https://doi.org/10.1007/s12044-015-0239-2

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