Open Access
Issue
ESAIM: M2AN
Volume 59, Number 5, September-October 2025
Page(s) 2613 - 2637
DOI https://doi.org/10.1051/m2an/2025063
Published online 17 September 2025
  1. T.P. Barrios, E.M. Behrens and M.A. Sánchez, A posteriori error analysis of an augmented mixed formulation in linear elasticity with mixed and dirichlet boundary conditions. Comput. Methods Appl. Mech. Eng. 200 (2011) 101–113. [Google Scholar]
  2. C. Bernardi, Optimal finite-element interpolation on curved domains. SIAM J. Numer. Anal. 26 (1989) 1212–1240. [CrossRef] [MathSciNet] [Google Scholar]
  3. A. Bonito and A. Demlow, Convergence and optimality of higher-order adaptive finite element methods for eigenvalue clusters. SIAM J. Numer. Anal. 54 (2016) 2379–2388. [Google Scholar]
  4. A. Bonito, A. Demlow and J. Owen, A priori error estimates for finite element approximations to eigenvalues and eigenfunctions of the Laplace-Beltrami operator. SIAM J. Numer. Anal. 56 (2018) 2963–2988. [Google Scholar]
  5. V. Bonnaillie-Noël, D. Brancherie, M. Dambrine, F. Hérau, S. Tordeux and G. Vial, Multiscale expansion and numerical approximation for surface defects, in CANUM 2010, 40e Congrès National d’Analyse Numérique. Vol. 33 of ESAIM Proc. EDP Sciences, Les Ulis, (2011) 22–35. [Google Scholar]
  6. S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods. Vol. 15. Springer, New York, NY (2002). [Google Scholar]
  7. C. Carstensen and G. Dolzmann, A posteriori error estimates for mixed fem in elasticity. Numer. Math. 81 (1998) 187–209. [Google Scholar]
  8. F. Caubet, J. Ghantous and C. Pierre, Numerical study of a diffusion equation with ventcel boundary condition using curved meshes. Monografías Matemáticas García de Galdeano. Preprint arXiv:2302.02680 (2023). [Google Scholar]
  9. F. Caubet, J. Ghantous and C. Pierre, Finite element analysis of a spectral problem on curved meshes occurring in diffusion with high order boundary conditions. Preprint arXiv:2404.13994 (2024). [Google Scholar]
  10. F. Caubet, J. Ghantous and C. Pierre, A priori error estimates of a poisson equation with ventcel boundary conditions on curved meshes. SIAM J. Numer. Anal. 62 (2024) 1929–1955. [Google Scholar]
  11. P.G. Ciarlet, Mathematical Elasticity. Vol III. Theory of Shells (2000). [Google Scholar]
  12. P.G. Ciarlet, The Finite Element Method for Elliptic Problems. Vol. 40 of Classics in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2002). [Google Scholar]
  13. P. Ciarlet, Mathematical Elasticity, Volume I: Three-Dimensional Elasticity. Classics in Applied Mathematics. Society for Industrial and Applied Mathematics (2022). [Google Scholar]
  14. P.G. Ciarlet and P.-A. Raviart, Interpolation theory over curved elements, with applications to finite element methods. Comp. Meth. Appl. Mech. Eng. 1 (1972) 217–249. [Google Scholar]
  15. C. Dapogny and P. Frey, Computation of the signed distance function to a discrete contour on adapted triangulation. Calcolo 49 (2012) 193–219. [Google Scholar]
  16. A. Demlow, Higher-order finite element methods and pointwise error estimates for elliptic problems on surfaces. SIAM J. Numer. Anal. 47 (2009) 805–827. [Google Scholar]
  17. A. Demlow and G. Dziuk, An adaptive finite element method for the Laplace–Beltrami operator on implicitly defined surfaces. SIAM J. Numer. Anal. 45 (2007) 421–442. [CrossRef] [MathSciNet] [Google Scholar]
  18. F. Dubois, Discrete vector potential representation of a divergence-free vector field in three-dimensional domains: numerical analysis of a model problem. SIAM J. Numer. Anal. 27 (1990) 1103–1141. [Google Scholar]
  19. G. Duvaut and J.L. Lions, Inequalities in Mechanics and Physics. Vol. 219. Springer Science & Business Media (2012). [Google Scholar]
  20. G. Dziuk, Finite elements for the Beltrami operator on arbitrary surfaces, in Partial Differential Equations and Calculus of Variations. Vol. 1357 of Lecture Notes in Math. Springer, Berlin (1988) 142–155. [Google Scholar]
  21. D. Edelmann, Isoparametric finite element analysis of a generalized Robin boundary value problem on curved domains. SMAI J. Comput. Math. 7 (2021) 57–73. [Google Scholar]
  22. C.M. Elliott and T. Ranner, Finite element analysis for a coupled bulk-surface partial differential equation. IMA J. Numer. Anal. 33 (2013) 377–402. [CrossRef] [MathSciNet] [Google Scholar]
  23. A. Ern and J.-L. Guermond, Theory and Practice of Finite Elements. Vol. 159 of Applied Mathematical Sciences. Springer-Verlag, New York (2004). [Google Scholar]
  24. K. Feng and Z.-C. Shi, Mathematical Theory of Elastic Structures. Springer-Verlag, Berlin; Science Press Beijing, Beijing (1996). [Google Scholar]
  25. G.N. Gatica and L.F. Gatica, On the a priori and a posteriori error analysis of a two-fold saddle-point approach for nonlinear incompressible elasticity. Int. J. Numer. Methods Eng. 68 (2006) 861–892. [Google Scholar]
  26. J. Ghantous, Consideration of high-order boundary conditions and numerical analysis of diffusion problems on curved meshes using high-order finite elements. Ph.D. thesis, Université de Pau et des Pays de l’Adour (UPPA) (2024). [Google Scholar]
  27. D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order. Classics in Mathematics. Springer-Verlag, Berlin (2001). Reprint of the 1998 edition. [Google Scholar]
  28. G.R. Goldstein, Derivation and physical interpretation of general boundary conditions. Adv. Differ. Equ. 11 (2006) 457–480. [Google Scholar]
  29. P. Grisvard, Elliptic Problems in Nonsmooth Domains. Vol. 69 of Classics in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2011). [Google Scholar]
  30. H. Haddar, Modèles asymptotiques en ferromagnétisme: couches minces et homogénéisation. Ph.D. thesis, Ecole des Ponts ParisTech (2000). [Google Scholar]
  31. P. Hansbo, M.G. Larson and K. Larsson, Analysis of finite element methods for vector laplacians on surfaces. IMA J. Numer. Anal. 40 (2020) 1652–1701. [Google Scholar]
  32. T. Kashiwabara, C.M. Colciago, L. Dedè and A. Quarteroni, Well-posedness, regularity, and convergence analysis of the finite element approximation of a generalized Robin boundary value problem. SIAM J. Numer. Anal. 53 (2015) 105–126. [Google Scholar]
  33. M. Lenoir, Optimal isoparametric finite elements and error estimates for domains involving curved boundaries. SIAM J. Numer. Anal. 23 (1986) 562–580. [Google Scholar]
  34. D. Mora and G. Rivera, A priori and a posteriori error estimates for a virtual element spectral analysis for the elasticity equations. IMA J. Numer. Anal. 40 (2020) 322–357. [Google Scholar]
  35. J.-C. Nédélec, Curved finite element methods for the solution of singular integral equations on surfaces in R3. Comput. Methods Appl. Mech. Eng. 8 (1976) 61–80. [Google Scholar]
  36. C. Pierre, The finite element library Cumin, curved meshes in numerical simulations. https://plmlab.math.cnrs.fr/cpierre1/cuminhal-0393713(v1) (2023). [Google Scholar]
  37. R. Rannacher, On finite element approximation of general boundary value problems in nonlinear elasticity. Calcolo 17 (1980) 175–193. [CrossRef] [MathSciNet] [Google Scholar]
  38. L.R. Scott, Finite element techniques for curved boundaries. Ph.D. thesis, Massachusetts Institute of Technology, ProQuest LLC, Ann Arbor, MI (1973). [Google Scholar]
  39. R. Scott, Interpolated boundary conditions in the finite element method. SIAM J. Numer. Anal. 12 (1975) 404–427. [Google Scholar]
  40. J.R. Stewart and T.J. Hughes, A tutorial in elementary finite element error analysis: a systematic presentation of a priori and a posteriori error estimates. Comput. Methods Appl. Mech. Eng. 158 (1998) 1–22. [Google Scholar]
  41. A.D. Ventcel, Semigroups of operators that correspond to a generalized differential operator of second order. Dokl. Akad. Nauk SSSR (N.S.) 111 (1956) 269–272. [Google Scholar]
  42. A.D. Ventcel, On boundary conditions for multi-dimensional diffusion processes. Theor. Probab. App. 4 (1959) 164–177. [Google Scholar]
  43. G. Vial, Analyse asymptotique multi-échelle et conditions aux limites approchées pour un problème de couche mince dans un domaine à coin. Ph.D. thesis, Université Rennes 1 (2003). [Google Scholar]

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