Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter May 7, 2019

The Discrete Steklov–Poincaré Operator Using Algebraic Dual Polynomials

  • Yi Zhang ORCID logo EMAIL logo , Varun Jain , Artur Palha and Marc Gerritsma

Abstract

In this paper, we will use algebraic dual polynomials to set up a discrete Steklov–Poincaré operator for the mixed formulation of the Poisson problem. The method will be applied in curvilinear coordinates and to a test problem which contains a singularity. Exponential convergence of the trace variable in H1/2-norm will be shown.

References

[1] V. I. Agoshkov, Poincaré–Steklov’s operators and domain decomposition methods in finite-dimensional spaces, First International Symposium on Domain Decomposition Methods for Partial Differential Equations (Paris 1987), SIAM, Philadelphia (1988), 73–112. Search in Google Scholar

[2] D. N. Arnold, F. Brezzi, B. Cockburn and L. D. Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal. 39 (2001/02), no. 5, 1749–1779. 10.1137/S0036142901384162Search in Google Scholar

[3] I. Babuška, J. T. Oden and J. K. Lee, Mixed-hybrid finite element approximations of second-order elliptic boundary-value problems, Comput. Methods Appl. Mech. Engrg. 11 (1977), no. 2, 175–206. 10.1016/0045-7825(78)90010-5Search in Google Scholar

[4] P. Bochev and M. Gerritsma, A spectral mimetic least-squares method, Comput. Math. Appl. 68 (2014), no. 11, 1480–1502. 10.1016/j.camwa.2014.09.014Search in Google Scholar

[5] D. Boffi, F. Brezzi and M. Fortin, Mixed Finite Element Methods and Applications, Springer Ser. Comput. Math. 44, Springer, Heidelberg, 2013. 10.1007/978-3-642-36519-5Search in Google Scholar

[6] C. Carstensen, L. Demkowicz and J. Gopalakrishnan, Breaking spaces and forms for the DPG method and applications including Maxwell equations, Comput. Math. Appl. 72 (2016), no. 3, 494–522. 10.1016/j.camwa.2016.05.004Search in Google Scholar

[7] B. Cockburn and J. Gopalakrishnan, A characterization of hybridized mixed methods for second order elliptic problems, SIAM J. Numer. Anal. 42 (2004), no. 1, 283–301. 10.1137/S0036142902417893Search in Google Scholar

[8] B. Cockburn, J. Gopalakrishnan and R. Lazarov, Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems, SIAM J. Numer. Anal. 47 (2009), no. 2, 1319–1365. 10.1137/070706616Search in Google Scholar

[9] D. E. Crabtree and E. V. Haynsworth, An identity for the Schur complement of a matrix, Proc. Amer. Math. Soc. 22 (1969), 364–366. 10.1090/S0002-9939-1969-0255573-1Search in Google Scholar

[10] D. De Klerk, D. Rixen and S. Voormeeren, General framework for dynamic substructuring: History, review, and classification of techniques, AIAA Journal 46 (2008), no. 5, 1169–1181. 10.2514/1.33274Search in Google Scholar

[11] L. Demkowicz and J. Gopalakrishnan, A class of discontinuous Petrov–Galerkin methods. Part I: The transport equation, Comput. Methods Appl. Mech. Engrg. 199 (2010), no. 23–24, 1558–1572. 10.1016/j.cma.2010.01.003Search in Google Scholar

[12] L. Demkowicz and J. Gopalakrishnan, A class of discontinuous Petrov-Galerkin methods. II. Optimal test functions, Numer. Methods Partial Differential Equations 27 (2011), no. 1, 70–105. 10.1002/num.20640Search in Google Scholar

[13] L. Demkowicz and J. Gopalakrishnan, Analysis of the DPG method for the Poisson equation, SIAM J. Numer. Anal. 49 (2011), no. 5, 1788–1809. 10.1137/100809799Search in Google Scholar

[14] B. Fraeijs de Veubeke, Displacement and equilibrium models in the finite element method, Internat. J. Numer. Methods Engrg. 52 (2001), no. 3, 287–342. Search in Google Scholar

[15] M. Gerritsma, Edge functions for spectral element methods, Spectral and High Order Methods for Partial Differential Equations, Lect. Notes Comput. Sci. Eng. 76, Springer, Heidelberg (2011), 199–207. 10.1007/978-3-642-15337-2_17Search in Google Scholar

[16] E. V. Haynsworth, Reduction of a matrix using properties of the Schur complement, Linear Algebra Appl. 3 (1970), 23–29. 10.1016/0024-3795(70)90025-XSearch in Google Scholar

[17] V. Jain, Y. Zhang, A. Palha and M. Gerritsma, Construction and application of algebraic dual polynomial representations for finite element methods, (2017), https://arxiv.org/abs/1712.09472. Search in Google Scholar

[18] A. Klawonn, O. B. Widlund and M. Dryja, Dual-primal FETI methods for three-dimensional elliptic problems with heterogeneous coefficients, SIAM J. Numer. Anal. 40 (2002), no. 1, 159–179. 10.1137/S0036142901388081Search in Google Scholar

[19] J. Kreeft and M. Gerritsma, Mixed mimetic spectral element method for Stokes flow: A pointwise divergence-free solution, J. Comput. Phys. 240 (2013), 284–309. 10.1016/j.jcp.2012.10.043Search in Google Scholar

[20] J. Moitinho de Almeida and E. Maunder, Equilibrium Finite Element Formulations, John Wiley & Sons, New York, 2017. 10.1002/9781118925782Search in Google Scholar

[21] J. T. Oden and L. F. Demkowicz, Applied Functional Analysis, 2nd ed., CRC Press, Boca Raton, 2010. 10.1201/b17181Search in Google Scholar

[22] A. Palha, P. P. Rebelo, R. Hiemstra, J. Kreeft and M. Gerritsma, Physics-compatible discretization techniques on single and dual grids, with application to the Poisson equation of volume forms, J. Comput. Phys. 257 (2014), 1394–1422. 10.1016/j.jcp.2013.08.005Search in Google Scholar

[23] T. H. H. Pian and C.-C. Wu, Hybrid and Incompatible Finite Element Methods, CRC Ser. Mod. Mech. Math. 4, Chapman & Hall/CRC, Boca Raton, 2006. 10.1201/9780203487693Search in Google Scholar

[24] A. Quarteroni and A. Valli, Theory and application of Steklov–Poincaré operators for boundary-value problems, Applied and Industrial Mathematics (Venice 1989), Math. Appl. 56, Kluwer Academic, Dordrecht (1991), 179–203. 10.1007/978-94-009-1908-2_14Search in Google Scholar

[25] A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations, Springer Ser. Comput. Math. 23, Springer, Berlin, 1994. 10.1007/978-3-540-85268-1Search in Google Scholar

[26] E. Wilson, The static condensation algorithm, Internat. J. Numer. Methods Engrg. 8 (1974), 198–203. 10.1002/nme.1620080115Search in Google Scholar

Received: 2018-08-13
Accepted: 2019-03-05
Published Online: 2019-05-07
Published in Print: 2019-07-01

© 2019 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 5.5.2024 from https://www.degruyter.com/document/doi/10.1515/cmam-2018-0208/html
Scroll to top button