A stabilized nonconforming finite element method for the elliptic Cauchy problem

Burman, Erik

doi:10.1090/mcom/3092

A stabilized nonconforming finite element method for the elliptic Cauchy problem
HTML articles powered by AMS MathViewer

by Erik Burman;

Math. Comp. 86 (2017), 75-96

DOI: https://doi.org/10.1090/mcom/3092

Published electronically: April 4, 2016

PDF | Request permission

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

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
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, DOI 10.1088/0266-5611/22/1/007
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, DOI 10.1088/0266-5611/22/4/012
Faker Ben Belgacem, Why is the Cauchy problem severely ill-posed?, Inverse Problems 23 (2007), no. 2, 823–836. MR 2309677, DOI 10.1088/0266-5611/23/2/020
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, DOI 10.1088/0266-5611/21/3/018
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, DOI 10.1088/0266-5611/22/2/002
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 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
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, DOI 10.1088/0266-5611/22/4/005
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
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, DOI 10.1137/120895123
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
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, DOI 10.1090/S0025-5718-1986-0842126-5
J. Hadamard, Sur les problèmes aux derivées partielles et leur signification physique, Bull. Univ. Princeton (1902).
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, DOI 10.4208/cicp.200110.060910a
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, DOI 10.1088/0266-5611/23/6/008
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
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
R. Lattès and J.-L. Lions, The method of quasi-reversibility. Applications to partial differential equations, Modern Analytic and Computational Methods in Science and Mathematics, No. 18, American Elsevier Publishing Co., Inc., New York, 1969. Translated from the French edition and edited by Richard Bellman. MR 243746
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, DOI 10.1016/0168-9274(95)00055-Y
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, DOI 10.1137/S0036142997316955
Andrey N. Tikhonov and Vasiliy Y. Arsenin, Solutions of ill-posed problems, Scripta Series in Mathematics, V. H. Winston & Sons, Washington, DC; John Wiley & Sons, New York-Toronto-London, 1977. Translated from the Russian; Preface by translation editor Fritz John. MR 455365