Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

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

 
 

 

A stabilized nonconforming finite element method for the elliptic Cauchy problem


Author: Erik Burman
Journal: Math. Comp. 86 (2017), 75-96
MSC (2010): Primary 65N12, 65N15, 65N20, 65N21, 65N30
DOI: https://doi.org/10.1090/mcom/3092
Published electronically: April 4, 2016
MathSciNet review: 3557794
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we propose a nonconforming finite element method for the solution of the ill-posed elliptic Cauchy problem. The recently derived framework from previous works of the author is extended to include the case of a nonconforming approximation space. We show that the use of such a space allows us to reduce the amount of stabilization necessary for convergence, even in the case of ill-posed problems. We derive error estimates using conditional stability estimates in the $ L^2$-norm.


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 (2005d:65179), 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 (2010k:35517), https://doi.org/10.1088/0266-5611/25/12/123004
  • [3] S. Andrieux, T. N. Baranger, and A. Ben Abda, Solving Cauchy problems by minimizing an energy-like functional, Inverse Problems 22 (2006), no. 1, 115-133. MR 2194187 (2007b:35084), https://doi.org/10.1088/0266-5611/22/1/007
  • [4] Mejdi Azaïez, Faker Ben Belgacem, and Henda El Fekih, On Cauchy's problem. II. Completion, regularization and approximation, Inverse Problems 22 (2006), no. 4, 1307-1336. MR 2249467 (2008b:35043), https://doi.org/10.1088/0266-5611/22/4/012
  • [5] Faker Ben Belgacem, Why is the Cauchy problem severely ill-posed?, Inverse Problems 23 (2007), no. 2, 823-836. MR 2309677 (2008c:35331), https://doi.org/10.1088/0266-5611/23/2/020
  • [6] L. Bourgeois, A mixed formulation of quasi-reversibility to solve the Cauchy problem for Laplace's equation, Inverse Problems 21 (2005), no. 3, 1087-1104. MR 2146823 (2006b:35334), https://doi.org/10.1088/0266-5611/21/3/018
  • [7] L. Bourgeois, Convergence rates for the quasi-reversibility method to solve the Cauchy problem for Laplace's equation, Inverse Problems 22 (2006), no. 2, 413-430. MR 2216406 (2007d:35277), https://doi.org/10.1088/0266-5611/22/2/002
  • [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 (English, with English and French summaries). MR 3237821, https://doi.org/10.1016/j.crma.2014.06.008
  • [10] 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 (2006i:65190), https://doi.org/10.1002/num.20076
  • [11] A. Chakib and A. Nachaoui, Convergence analysis for finite element approximation to an inverse Cauchy problem, Inverse Problems 22 (2006), no. 4, 1191-1206. MR 2249460 (2007h:49040), https://doi.org/10.1088/0266-5611/22/4/005
  • [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 (49 #8401)
  • [13] Jérémi Dardé, Antti Hannukainen, and Nuutti Hyvönen, An $ H_{\mathsf {div}}$-based mixed quasi-reversibility method for solving elliptic Cauchy problems, SIAM J. Numer. Anal. 51 (2013), no. 4, 2123-2148. MR 3079321, https://doi.org/10.1137/120895123
  • [14] Robert Eymard, Thierry Gallouët, and Raphaèle Herbin, Error estimate for approximate solutions of a nonlinear convection-diffusion problem, Adv. Differential Equations 7 (2002), no. 4, 419-440. MR 1869118 (2002h:35156)
  • [15] R. S. Falk and P. B. Monk, Logarithmic convexity for discrete harmonic functions and the approximation of the Cauchy problem for Poisson's equation, Math. Comp. 47 (1986), no. 175, 135-149. MR 842126 (87j:65109), https://doi.org/10.2307/2008085
  • [16] J. Hadamard, Sur les problèmes aux derivées partielles et leur signification physique, Bull. Univ. Princeton (1902).
  • [17] Houde Han, Leevan Ling, and Tomoya Takeuchi, An energy regularization for Cauchy problems of Laplace equation in annulus domain, Commun. Comput. Phys. 9 (2011), no. 4, 878-896. MR 2734356, https://doi.org/10.4208/cicp.200110.060910a
  • [18] Weimin Han, Jianguo Huang, Kamran Kazmi, and Yu Chen, A numerical method for a Cauchy problem for elliptic partial differential equations, Inverse Problems 23 (2007), no. 6, 2401-2415. MR 2441010 (2009f:65138), https://doi.org/10.1088/0266-5611/23/6/008
  • [19] 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 (2003j:74057), https://doi.org/10.1016/S0045-7825(01)00358-9
  • [20] 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 (2004b:65184), https://doi.org/10.1051/m2an:2003020
  • [21] F. Hecht, New development in freefem++, J. Numer. Math. 20 (2012), no. 3-4, 251-265. MR 3043640
  • [22] 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 (electronic). MR 2034620 (2005d:65192), https://doi.org/10.1137/S0036142902405217
  • [23] R. Lattès and J.-L. Lions, The Method of Quasi-reversibility. Applications to Partial Differential Equations, Translated from the French edition and edited by Richard Bellman. Modern Analytic and Computational Methods in Science and Mathematics, No. 18, American Elsevier Publishing Co., Inc., New York, 1969. MR 0243746 (39 #5067)
  • [24] W. Lucht, A finite element method for an ill-posed problem, Appl. Numer. Math. 18 (1995), no. 1-3, 253-266. Seventh Conference on the Numerical Treatment of Differential Equations (Halle, 1994). MR 1357921 (96f:65154), https://doi.org/10.1016/0168-9274(95)00055-Y
  • [25] Hans-Jürgen Reinhardt, Houde Han, and Dinh Nho Hào, Stability and regularization of a discrete approximation to the Cauchy problem for Laplace's equation, SIAM J. Numer. Anal. 36 (1999), no. 3, 890-905. MR 1681021 (2000a:65166), https://doi.org/10.1137/S0036142997316955
  • [26] Andrey N. Tikhonov and Vasiliy Y. Arsenin, Solutions of Ill-posed Problems, V. H. Winston & Sons, Washington, D.C.: John Wiley & Sons, New York-Toronto, Ont.-London, 1977. Translated from the Russian; Preface by translation editor Fritz John; Scripta Series in Mathematics. MR 0455365 (56 #13604)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2010): 65N12, 65N15, 65N20, 65N21, 65N30

Retrieve articles in all journals with MSC (2010): 65N12, 65N15, 65N20, 65N21, 65N30


Additional Information

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

DOI: https://doi.org/10.1090/mcom/3092
Received by editor(s): June 17, 2014
Received by editor(s) in revised form: March 20, 2015, and June 8, 2015
Published electronically: April 4, 2016
Article copyright: © Copyright 2016 American Mathematical Society

American Mathematical Society