Skip to main content
SpringerLink
Log in

Analysis of Lavrentiev-finite element methods for data completion problems

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

Notes

  1. The symbol \(\alpha _+\) stand for any real number strictly larger than \(\alpha \).

References

  1. Adams, R.A., Fournier, J.J.: Sobolev Spaces. Pure and Applied Mathematics, vol. 140. Academic Press, New-York, London (2003)

    Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  4. Azaïez, M., Ben Belgacem, F., El Fekih, H.: On Cauchy’s problem. II. Completion, regularization and approximation. Inverse Prob. 22, 1307–1336 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  7. Ben Belgacem, F.: Why is the Cauchy’s problem severely ill-posed? Inverse Prob. 23, 823–836 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  9. Ben Belgacem, F., El Fekih, H.: On Cauchy’s problem. I. A variational Steklov–Poincaré theory. Inverse Prob. 21, 1915–1936 (2005)

    Article  MATH  Google Scholar 

  10. Ben Belgacem, F., El Fekih, H., Jelassi, F.: The Lavrentiev regularization of the data completion problem. Inverse Prob. 24, 045009 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  11. Bernardi, C., Maday, Y., Rapetti, F.: Discrétisations variationnelles de problèmes aux limites elliptiques, vol. 45. Springer, Paris (2004)

    MATH  Google Scholar 

  12. Boukari, Y., Haddar, H.: A convergent data completion algorithm using surface integral equations. Inverse Prob. 31, 035011 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  19. Ciarlet Jr., P.: Analysis of the Scott–Zhang interpolation in the fractional order Sobolev spaces. J. Numer. Math. 21, 173–180 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  22. Dauge, M.: Elliptic Boundary Value Problems on Corner Domains: Smoothness and Asymptotic of Solutions, vol. 1341. Springer, Berlin (1988)

    MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  27. Grisvard, P.: Elliptic Problems in Nonsmooth Domains. vol. CL69 of Classics in Applied Mathematics. SIAM, (2011)

  28. Hadarmard, J.: Lectures on Cauchy’s Problem in Linear Partial Differential Equation. Dover, New York (1953)

    Google Scholar 

  29. Isakov, V.: Inverse Problems for Partial Differential Equations. Vol. 127 of Applied Mathematical Sciences. Springer, New York (2006)

    MATH  Google Scholar 

  30. Jerison, D.S., Kenig, C.E.: The Neumann problem on Lipschitz domains. Bull. Am. Math. Soc. (N. S.) 4, 203–207 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  31. Jerison, D.S., Kenig, C.E.: The Inhomogeneous Dirichlet problem in Lipschitz domains. J. Funct. Anal. 130, 161–219 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  32. Klibanov, M.V.: Carleman estimates for the regularization of ill-posed Cauchy problems. J. Appl. Numer. Math. 94, 46–74 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  33. Kohn, R.V., Vogelius, M.S.: Determining conductivity by boundary measurements II. Interior results. Commun. Pure Appl. Math. 38, 643–667 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  35. Nitsche, J.A., Schatz, A.H.: Interior estimates for Ritz–Galerkin methods. Math. Comput. 28, 937–958 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  36. Quarteroni, A., Valli, A.: Domain Decomposition Methods for Partial Differential Equations. Oxford Science Publications, Oxford (1999)

    MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  38. Scott, L.R., Zhang, S.: Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput. 54, 483–493 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  40. Strang, G., Fix, G.: An Analysis of the Finite Element Method. Cambridge Press, Wellesley (2008)

    MATH  Google Scholar 

  41. Tataru, D.: A-priori estimates of Carleman’s type in domains with boundary. Journal des Mathematiques Pures et Appliquées 73, 355–387 (1994)

    MathSciNet  MATH  Google Scholar 

  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)

Download references

Author information

Authors and Affiliations

  1. UTC, EA 2222, Laboratoire de Mathématiques Appliquées de Compiègne, Sorbonne Universités, 60205, Compiegne, France

    F. Ben Belgacem & F. Jelassi

  2. UPMC, UMR-CNRS 7598, Laboratoire Jacques-Louis Lions, Sorbonne Universités, 75005, Paris, France

    V. Girault

Authors
  1. F. Ben Belgacem
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. V. Girault
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. F. Jelassi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to F. Ben Belgacem.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Belgacem, F.B., Girault, V. & Jelassi, F. Analysis of Lavrentiev-finite element methods for data completion problems. Numer. Math. 139, 1–25 (2018). https://doi.org/10.1007/s00211-017-0930-6

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00211-017-0930-6

Mathematics Subject Classification

Search

Navigation