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The Discrete Steklov–Poincaré Operator Using Algebraic Dual Polynomials

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


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.


[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), 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/ 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/ 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

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