Stabilized nonconforming finite element methods for data assimilation in incompressible flows

Burman, Erik; Hansbo, Peter

doi:10.1090/mcom/3255

Stabilized nonconforming finite element methods for data assimilation in incompressible flows
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by Erik Burman and Peter Hansbo;

Math. Comp. 87 (2018), 1029-1050

DOI: https://doi.org/10.1090/mcom/3255

Published electronically: September 19, 2017

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Abstract:

We consider a stabilized nonconforming finite element method for data assimilation in incompressible flow subject to the Stokes equations. The method uses a primal dual structure that allows for the inclusion of nonstandard data. Error estimates are obtained that are optimal compared to the conditional stability of the ill-posed data assimilation problem.

References

Y. Achdou, C. Bernardi, and F. Coquel, A priori and a posteriori analysis of finite volume discretizations of Darcy’s equations, Numer. Math. 96 (2003), no. 1, 17–42. MR 2018789, DOI 10.1007/s00211-002-0436-7
Giovanni Alessandrini, Luca Rondi, Edi Rosset, and Sergio Vessella, The stability for the Cauchy problem for elliptic equations, Inverse Problems 25 (2009), no. 12, 123004, 47. MR 2565570, DOI 10.1088/0266-5611/25/12/123004
Mehdi Badra, Fabien Caubet, and Jérémi Dardé, Stability estimates for Navier-Stokes equations and application to inverse problems, Discrete Contin. Dyn. Syst. Ser. B 21 (2016), no. 8, 2379–2407. MR 3555120, DOI 10.3934/dcdsb.2016052
A. Ben Abda, I. B. Saad, and M. Hassine, Data completion for the Stokes system, Comptes Rendus Mecanique 337 (2009), no. 9–10, 703–708.
Daniele Boffi, Franco Brezzi, and Michel Fortin, Mixed finite element methods and applications, Springer Series in Computational Mathematics, vol. 44, Springer, Heidelberg, 2013. MR 3097958, DOI 10.1007/978-3-642-36519-5
Muriel Boulakia, Anne-Claire Egloffe, and Céline Grandmont, Stability estimates for the unique continuation property of the Stokes system and for an inverse boundary coefficient problem, Inverse Problems 29 (2013), no. 11, 115001, 21. MR 3116337, DOI 10.1088/0266-5611/29/11/115001
Laurent Bourgeois and Jérémi Dardé, The “exterior approach” to solve the inverse obstacle problem for the Stokes system, Inverse Probl. Imaging 8 (2014), no. 1, 23–51. MR 3180411, DOI 10.3934/ipi.2014.8.23
Erik Burman, Stabilized finite element methods for nonsymmetric, noncoercive, and ill-posed problems. Part I: Elliptic equations, SIAM J. Sci. Comput. 35 (2013), no. 6, A2752–A2780. MR 3134434, DOI 10.1137/130916862
Erik Burman, Error estimates for stabilized finite element methods applied to ill-posed problems, C. R. Math. Acad. Sci. Paris 352 (2014), no. 7-8, 655–659 (English, with English and French summaries). MR 3237821, DOI 10.1016/j.crma.2014.06.008
Erik Burman, A stabilized nonconforming finite element method for the elliptic Cauchy problem, Math. Comp. 86 (2017), no. 303, 75–96. MR 3557794, DOI 10.1090/mcom/3092
Erik Burman and Peter Hansbo, Stabilized Crouzeix-Raviart element for the Darcy-Stokes problem, Numer. Methods Partial Differential Equations 21 (2005), no. 5, 986–997. MR 2154230, DOI 10.1002/num.20076
M. Crouzeix and P.-A. Raviart, Conforming and nonconforming finite element methods for solving the stationary Stokes equations. I, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge 7 (1973), no. R-3, 33–75. MR 343661
Daniele Antonio Di Pietro and Alexandre Ern, Mathematical aspects of discontinuous Galerkin methods, Mathématiques & Applications (Berlin) [Mathematics & Applications], vol. 69, Springer, Heidelberg, 2012. MR 2882148, DOI 10.1007/978-3-642-22980-0
Alexandre Ern and Jean-Luc Guermond, Theory and practice of finite elements, Applied Mathematical Sciences, vol. 159, Springer-Verlag, New York, 2004. MR 2050138, DOI 10.1007/978-1-4757-4355-5
Caroline Fabre and Gilles Lebeau, Prolongement unique des solutions de l’equation de Stokes, Comm. Partial Differential Equations 21 (1996), no. 3-4, 573–596 (French, with English and French summaries). MR 1387461, DOI 10.1080/03605309608821198
Vivette Girault and Pierre-Arnaud Raviart, Finite element methods for Navier-Stokes equations, Springer Series in Computational Mathematics, vol. 5, Springer-Verlag, Berlin, 1986. Theory and algorithms. MR 851383, DOI 10.1007/978-3-642-61623-5
Peter Hansbo and Mats G. Larson, Discontinuous Galerkin methods for incompressible and nearly incompressible elasticity by Nitsche’s method, Comput. Methods Appl. Mech. Engrg. 191 (2002), no. 17-18, 1895–1908. MR 1886000, DOI 10.1016/S0045-7825(01)00358-9
Peter Hansbo and Mats G. Larson, Discontinuous Galerkin and the Crouzeix-Raviart element: application to elasticity, M2AN Math. Model. Numer. Anal. 37 (2003), no. 1, 63–72. MR 1972650, DOI 10.1051/m2an:2003020
F. Hecht, New development in freefem++, J. Numer. Math. 20 (2012), no. 3-4, 251–265. MR 3043640, DOI 10.1515/jnum-2012-0013
Ronald H. W. Hoppe and Barbara Wohlmuth, Element-oriented and edge-oriented local error estimators for nonconforming finite element methods, RAIRO Modél. Math. Anal. Numér. 30 (1996), no. 2, 237–263 (English, with English and French summaries). MR 1382112, DOI 10.1051/m2an/1996300202371
Ohannes A. Karakashian and Frederic Pascal, A posteriori error estimates for a discontinuous Galerkin approximation of second-order elliptic problems, SIAM J. Numer. Anal. 41 (2003), no. 6, 2374–2399. MR 2034620, DOI 10.1137/S0036142902405217
Ching-Lung Lin, Gunther Uhlmann, and Jenn-Nan Wang, Optimal three-ball inequalities and quantitative uniqueness for the Stokes system, Discrete Contin. Dyn. Syst. 28 (2010), no. 3, 1273–1290. MR 2644789, DOI 10.3934/dcds.2010.28.1273
G. Seregin, Lecture notes on regularity theory for the Navier-Stokes equations, Oxford University, 2014.
C. Wang and J. Wang, A primal-dual weak Galerkin finite element method for second order elliptic equations in non-divergence form, Math. Comp., electronically published on June 27, 2017, DOI: https://doi.org/10.1090/mcom/3220 (to appear in print).