Numerische Mathematik

, Volume 139, Issue 1, pp 1–25 | Cite as

Analysis of Lavrentiev-finite element methods for data completion problems

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Abstract

The variational finite element solution of Cauchy’s problem, expressed in the Steklov–Poincaré framework and regularized by the Lavrentiev method, has been introduced and computationally assessed in Azaïez et al. (Inverse Probl Sci Eng 18:1063–1086, 2011). The present work concentrates on the numerical analysis of the semi-discrete problem. We perform the mathematical study of the error to rigorously establish the convergence of the global bias-variance error.

Mathematics Subject Classification

65L09 65F22 65L60 

References

  1. 1.
    Adams, R.A., Fournier, J.J.: Sobolev Spaces. Pure and Applied Mathematics, vol. 140. Academic Press, New-York, London (2003)Google Scholar
  2. 2.
    Andrieux, S., Baranger, T.N.: On the determination of missing boundary data for solids with nonlinear material behaviors, using displacement fields measured on a part of their boundaries. J. Mech. Phys. Solids 97, 140–155 (2016)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Azaïez, M., Ben Belgacem, F., Du, D.T., Jelassi, F.: A finite element model for the data completion problem: analysis and assessment. Inverse Prob. Sci. Eng. 18, 1063–1086 (2011)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Azaïez, M., Ben Belgacem, F., El Fekih, H.: On Cauchy’s problem. II. Completion, regularization and approximation. Inverse Prob. 22, 1307–1336 (2006)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Babuška, I., Suri, M.: The \(h\)-\(p\) version of the finite element method with quasiuniform meshes. RAIRO, Modélisation Mathématique et Analyse Numérique 21, 199–238 (1987)MathSciNetMATHGoogle Scholar
  6. 6.
    Bank, R.E., Yserentant, H.: On the \(H^1\)-stability of the \(L^2\)-projection onto finite element spaces. Numer. Math. 126, 361–381 (2014)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Ben Belgacem, F.: Why is the Cauchy’s problem severely ill-posed? Inverse Prob. 23, 823–836 (2007)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Ben Belgacem, F., Du, D.T., Jelassi, F.: Local convergence of the Lavrentiev method for the Cauchy problem via a Carleman inequality. J. Sci. Comput. 53, 320–341 (2012)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Ben Belgacem, F., El Fekih, H.: On Cauchy’s problem. I. A variational Steklov–Poincaré theory. Inverse Prob. 21, 1915–1936 (2005)CrossRefMATHGoogle Scholar
  10. 10.
    Ben Belgacem, F., El Fekih, H., Jelassi, F.: The Lavrentiev regularization of the data completion problem. Inverse Prob. 24, 045009 (2008)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Bernardi, C., Maday, Y., Rapetti, F.: Discrétisations variationnelles de problèmes aux limites elliptiques, vol. 45. Springer, Paris (2004)MATHGoogle Scholar
  12. 12.
    Boukari, Y., Haddar, H.: A convergent data completion algorithm using surface integral equations. Inverse Prob. 31, 035011 (2015)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Bourgeois, L.: Convergence rates for the quasi-reversibility method to solve the Cauchy problem for Laplace’s equation. Inverse Prob. 22, 413–430 (2006)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Bourgeois, L., Dardé, J.: A duality-based method of quasi-reversibility to solve the Cauchy problem in the presence of noisy data. Inverse Prob. 26, 095016 (2010)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Bramble, J.H., Pasciak, J.E., Steinbach, O.: On the stability of the \(L^2\) projection in \(H^1(\Omega )\). Math. Comput. 71, 147–156 (2002)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Cao, H., Klibanov, M.V., Pereverzev, S.V.: A Carleman estimate and the balancing principle in the quasi-reversibility method for solving the Cauchy problem for the Laplace equation. Inverse Prob. 25, 035005 (2009)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Cheng, X.L., Gong, R.F., Han, W.: A coupled complex boundary method for the Cauchy problem. Inverse Prob. Sci. Eng. 24, 1510–1527 (2016)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Ciarlet, P.G.: Basic error estimates for elliptic problems. In: Ciarlet, P.-G., Lions, J.-L. (eds.) Handbook of Numerical Analysis, vol. II, pp. 17–351. II, North-Holland, Amsterdam, Handb. Numer. Anal. (1991)Google Scholar
  19. 19.
    Ciarlet Jr., P.: Analysis of the Scott–Zhang interpolation in the fractional order Sobolev spaces. J. Numer. Math. 21, 173–180 (2013)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Crouzeix, M., Thomée, V.: The stability in \(L^p\) and \( W^{1, p}\) of the \(L^2\)-projection on finite element function spaces. Math. Comput. 48, 521–532 (1987)MATHGoogle Scholar
  21. 21.
    Dardé, J., Hannukainen, A., Hyvönen, N.: An \(H_{{\rm div}}\)-based mixed quasi-reversibility method for solving elliptic Cauchy problems. SIAM J. Numer. Anal. 51, 2123–2148 (2013)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Dauge, M.: Elliptic Boundary Value Problems on Corner Domains: Smoothness and Asymptotic of Solutions, vol. 1341. Springer, Berlin (1988)MATHGoogle Scholar
  23. 23.
    Delvare, D., Cimetière, A.: A robust data completion method for two dimensional Cauchy problems associated with the Laplace equation. Eur. J. Comput. Mech. 20, 309–340 (2011)CrossRefGoogle Scholar
  24. 24.
    Du, D.T.: A Lavrentiev Finite Element Model for the Cauchy Problem of Data Completion: Analysis and Numerical Assessment, PhD thesis, Université de Technologie de Compiègne, Compiègne, (2011)Google Scholar
  25. 25.
    Feng, X., Eldén, L.: Solving a Cauchy problem for a 3D elliptic PDE with variable coefficients by a quasi-boundary-value method. Inverse Prob. 30, 015005 (2014)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Girault, V., Lions, J.-L.: Two-grid finite-element scheme for the transient Navier–Stokes problem. Modél. Math. Anal. Numér. 35, 945–980 (2001)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Grisvard, P.: Elliptic Problems in Nonsmooth Domains. vol. CL69 of Classics in Applied Mathematics. SIAM, (2011)Google Scholar
  28. 28.
    Hadarmard, J.: Lectures on Cauchy’s Problem in Linear Partial Differential Equation. Dover, New York (1953)Google Scholar
  29. 29.
    Isakov, V.: Inverse Problems for Partial Differential Equations. Vol. 127 of Applied Mathematical Sciences. Springer, New York (2006)MATHGoogle Scholar
  30. 30.
    Jerison, D.S., Kenig, C.E.: The Neumann problem on Lipschitz domains. Bull. Am. Math. Soc. (N. S.) 4, 203–207 (1981)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Jerison, D.S., Kenig, C.E.: The Inhomogeneous Dirichlet problem in Lipschitz domains. J. Funct. Anal. 130, 161–219 (1995)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Klibanov, M.V.: Carleman estimates for the regularization of ill-posed Cauchy problems. J. Appl. Numer. Math. 94, 46–74 (2015)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Kohn, R.V., Vogelius, M.S.: Determining conductivity by boundary measurements II. Interior results. Commun. Pure Appl. Math. 38, 643–667 (1985)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Marin, L., Lesnic, D.: The method of fundamental solutions for the Cauchy problem in two-dimensional linear elasticity. Int. J. Solids Struct. 41, 3425–3438 (2004)CrossRefMATHGoogle Scholar
  35. 35.
    Nitsche, J.A., Schatz, A.H.: Interior estimates for Ritz–Galerkin methods. Math. Comput. 28, 937–958 (1974)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Quarteroni, A., Valli, A.: Domain Decomposition Methods for Partial Differential Equations. Oxford Science Publications, Oxford (1999)MATHGoogle Scholar
  37. 37.
    Rischette, R., Baranger, T.N., Débit, N.: Numerical analysis of an energy-like minimization method to solve the Cauchy problem with noisy data. J. Comput. Appl. Math. 235, 3257–3269 (2011)MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Scott, L.R., Zhang, S.: Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput. 54, 483–493 (1990)MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Shigeta, T., Young, D.L.: Method of fundamental solutions with optimal regularization techniques for the Cauchy problem of the Laplace equation with singular points. J. Comput. Phys. 228, 1903–1915 (2009)MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    Strang, G., Fix, G.: An Analysis of the Finite Element Method. Cambridge Press, Wellesley (2008)MATHGoogle Scholar
  41. 41.
    Tataru, D.: A-priori estimates of Carleman’s type in domains with boundary. Journal des Mathematiques Pures et Appliquées 73, 355–387 (1994)MathSciNetMATHGoogle Scholar
  42. 42.
    Wahlbin, L.B.: Local behavior in finite element methods. In: Ciarlet, P.-G., Lions, J.-L. (eds.) Handbook of Numerical Analysis, vol. II, pp. 353–522. II, Numer. Anal. North-Holland, Amsterdam, Handb. (1991)Google Scholar

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© Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.UTC, EA 2222, Laboratoire de Mathématiques Appliquées de CompiègneSorbonne UniversitésCompiegneFrance
  2. 2.UPMC, UMR-CNRS 7598, Laboratoire Jacques-Louis LionsSorbonne UniversitésParisFrance

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