Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

Stabilized nonconforming finite element methods for data assimilation in incompressible flows


Authors: Erik Burman and Peter Hansbo
Journal: Math. Comp. 87 (2018), 1029-1050
MSC (2010): Primary 65N30, 65N20; Secondary 65N12, 76D07, 76D55
DOI: https://doi.org/10.1090/mcom/3255
Published electronically: September 19, 2017
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [1] 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, https://doi.org/10.1007/s00211-002-0436-7
  • [2] 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, https://doi.org/10.1088/0266-5611/25/12/123004
  • [3] 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, https://doi.org/10.3934/dcdsb.2016052
  • [4] 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.
  • [5] 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
  • [6] 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, https://doi.org/10.1088/0266-5611/29/11/115001
  • [7] 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, https://doi.org/10.3934/ipi.2014.8.23
  • [8] 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, https://doi.org/10.1137/130916862
  • [9] 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. MR 3237821, https://doi.org/10.1016/j.crma.2014.06.008
  • [10] Erik Burman, A stabilized nonconforming finite element method for the elliptic Cauchy problem, Math. Comp. 86 (2017), no. 303, 75-96. MR 3557794, https://doi.org/10.1090/mcom/3092
  • [11] 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, https://doi.org/10.1002/num.20076
  • [12] 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 0343661
  • [13] 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
  • [14] Alexandre Ern and Jean-Luc Guermond, Theory and Practice of Finite Elements, Applied Mathematical Sciences, vol. 159, Springer-Verlag, New York, 2004. MR 2050138
  • [15] 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. MR 1387461, https://doi.org/10.1080/03605309608821198
  • [16] 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
  • [17] 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, https://doi.org/10.1016/S0045-7825(01)00358-9
  • [18] 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, https://doi.org/10.1051/m2an:2003020
  • [19] F. Hecht, New development in freefem++, J. Numer. Math. 20 (2012), no. 3-4, 251-265. MR 3043640
  • [20] 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. MR 1382112
  • [21] 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, https://doi.org/10.1137/S0036142902405217
  • [22] 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, https://doi.org/10.3934/dcds.2010.28.1273
  • [23] G. Seregin, Lecture notes on regularity theory for the Navier-Stokes equations, Oxford University, 2014.
  • [24] 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).

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2010): 65N30, 65N20, 65N12, 76D07, 76D55

Retrieve articles in all journals with MSC (2010): 65N30, 65N20, 65N12, 76D07, 76D55


Additional Information

Erik Burman
Affiliation: Department of Mathematics, University College London, London, UK-WC1E 6BT, United Kingdom
Email: e.burman@ucl.ac.uk

Peter Hansbo
Affiliation: Department of Mechanical Engineering, Jönköping University, SE-55111 Jönköping, Sweden
Email: peter.hansbo@ju.se

DOI: https://doi.org/10.1090/mcom/3255
Received by editor(s): March 21, 2016
Received by editor(s) in revised form: November 14, 2016
Published electronically: September 19, 2017
Article copyright: © Copyright 2017 American Mathematical Society

American Mathematical Society