EDP Sciences logo
Subscriber Authentication Point
Search
Menu
All issues Volume 56 / No 6 (November-December 2022) ESAIM: M2AN, 56 6 (2022) 2051-2079 References
Open Access
Issue
ESAIM: M2AN
Volume 56, Number 6, November-December 2022
Page(s) 2051 - 2079
DOI https://doi.org/10.1051/m2an/2022058
Published online 14 September 2022
  1. C. Amrouche and N.E.H. Seloula, Lp-theory for vector potentials and Sobolev’s inequalities for vector fields: application to the Stokes equations with pressure boundary conditions. Math. Models Methods Appl. Sci. 23 (2013) 37–92. [Google Scholar]
  2. O. Axelsson and I. Gustafsson, An iterative solver for a mixed variable variational formulation of the (first) biharmonic problem. Comput. Methods Appl. Mech. Eng. 20 (1979) 9–16. [CrossRef] [Google Scholar]
  3. O. Axelsson and N. Munksgaard, A class of preconditioned conjugate gradient methods for the solution of a mixed finite element discretization of the biharmonic operator. Int. J. Numer. Methods Eng. 14 (1979) 1001–1019. [CrossRef] [Google Scholar]
  4. J.W. Barrett and W.B. Liu, Finite element approximation of the p-Laplacian. Math. Comput. 204 (1993) 523–537. [Google Scholar]
  5. S. Bartels, Error control and adaptivity for a variational model problem defined on functions of bounded variation. Math. Comput. 293 (2015) 1217–1240. [Google Scholar]
  6. F. Bernis, J. Garcia-Azorero and I. Peral, Existence and multiplicity of nontrivial solutions in semilinear critical problems of fourth order. Adv. Differ. Equ. 1 (1996) 219–240. [Google Scholar]
  7. J.F. Bonder and J.D. Rossi, A fourth order elliptic equation with nonlinear boundary conditions. Nonlinear Anal. 49 (2002) 1037–1047. [CrossRef] [MathSciNet] [Google Scholar]
  8. D. Braess, Finite Elements, Theory, Fast Solvers and Applications in Solid Mechanics, 3rd edition. Cambridge University Press, Cambridge (2007). [CrossRef] [Google Scholar]
  9. D. Braess, V. Pillwein and J. Schöberl, Equilibrated residual error estimates are p-robust. Comp. Meth. Applied Mech. Eng. 198 (2009) 1189–1197. [CrossRef] [Google Scholar]
  10. D. Braess, T. Fraunholz and R.H.W. Hoppe, An equilibrated a posteriori error estimator for the interior penalty discontinuous Galerkin method. SIAM J. Numer. Anal. 52 (2014) 2121–2136. [Google Scholar]
  11. D. Braess, R.H.W. Hoppe and C. Linsenmann, A two-energies principle for the biharmonic equation and an a posteriori error estimator for an interior penalty discontinuous Galerkin approximation. ESAIM: M2AN 52 (2019) 2479–2504. [Google Scholar]
  12. S.C. Brenner and L.-Y. Sung, C0 interior penalty methods for fourth order elliptic boundary value problems on polygonal domains. J. Sci. Comput. 2223 (2005) 83–118. [CrossRef] [Google Scholar]
  13. S.C. Brenner, T. Gudi and L.-Y. Sung, An a posteriori error estimator for a quadratic C0-interior penalty method for the biharmonic problem. IMA J. Numer. Anal. 30 (2010) 777–798. [CrossRef] [MathSciNet] [Google Scholar]
  14. F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer, Berlin-Heidelberg-New York (1991). [CrossRef] [Google Scholar]
  15. A. Buffa and C. Ortner, Compact embeddings of broken Sobolev spaces and applications. IMA J. Numer. Anal. 29 (2009) 827–855. [CrossRef] [MathSciNet] [Google Scholar]
  16. Q.H. Choi and T. Jung, Multiplicity of solutions and source terms in a fourth order nonlinear elliptic equations. Acta Math. Sci. 19 (1999) 316–374. [Google Scholar]
  17. P.G. Ciarlet, The Finite Element Method for Elliptic Problems. SIAM, Philadelphia (2002). [Google Scholar]
  18. P.G. Ciarlet and P.A. Raviart, A mixed finite element method for the biharmonic equation. In: Proceedings of a Symposium on Mathematical Aspects of Finite Elements in Partial Differential Equations, edited by C. de Boor. Academic Press, New York (1974) 125–145. [CrossRef] [Google Scholar]
  19. B. Cockburn, B. Dong and J. Guzman, A hybridizable and superconvergent discontinuous Galerkin method for biharmonic problems. J. Sci. Comput. 40 (2009) 141–187. [Google Scholar]
  20. M. Crouzeix and V. Thomée, The stability in Lp and Formula of the L2 projection onto finite element function spaces. Math. Comput. 48 (1987) 521–532. [Google Scholar]
  21. Y. Deng and G. Wang, On inhomogeneous biharmonic equations involving critical exponents. Proc. R. Soc. Edinburgh Sect. A 129 (1999) 925–946. [Google Scholar]
  22. L. Diening, D. Kröner, M. Ruzicka and I. Toulopoulos, A local discontinuous Galerkin approximation for systems with p-structure. IMA J. Numer. Anal. 34 (2013) 1447–1488. [Google Scholar]
  23. D.A. DiPietro and A. Ern, Mathematical Aspects of Discontinuous Galerkin Methods. Springer, Berlin Heidelberg-New York (2012). [CrossRef] [Google Scholar]
  24. M. Dobrowolski, On finite element methods for nonlinear elliptic problems with corners. In: Springer Lecture Notes in Mathematics. Vol. 1121. Berlin-Heidelberg-New York, New York, Heidelberg (1986) 85–103. [Google Scholar]
  25. W. Dörfler, A convergent adaptive algorithm for Poisson’s equation. SIAM J. Numer. Anal. 33 (1996) 1106–1124. [Google Scholar]
  26. M.R. Dostanić, Inequality of Poincaré-Friedrichs on Lp spaces. Matematički Vesnik 57 (2005) 11–14. [Google Scholar]
  27. J. Douglas Jr, T. Dupont, P. Percell and R. Scott, A family of C1 finite elements with optimal approximation properties for various Galerkin methods for 2nd and 4th order problems. RAIRO Anal. Numér. 13 (1979) 227–255. [CrossRef] [EDP Sciences] [MathSciNet] [Google Scholar]
  28. I. Ekeland and R. Temam, Convex Analysis and Variational Problems. SIAM, Philadelphia (1999). [Google Scholar]
  29. G. Engel, K. Garikipati, T.J.R. Hughes, M.G. Larson, L. Mazzei and R.L. Taylor, Continuous/discontinuous finite element approximations of fourth order elliptic problems in structural and continuum mechanics with applications to thin beams and plates, and strain gradient elasticity. Comput. Methods Appl. Mech. Eng. 191 (2002) 3669–3750. [CrossRef] [Google Scholar]
  30. E.H. Georgoulis and P. Houston, Discontinuous Galerkin methods for the biharmonic problem. IMA J. Numer. Anal. 29 (2009) 573–594. [CrossRef] [MathSciNet] [Google Scholar]
  31. E.H. Georgoulis, P. Houston and J. Virtanen, An a posteriori error indicator for discontinuous Galerkin approximations of fourth order elliptic problems. IMA J. Numer. Anal. 31 (2011) 281–298. [CrossRef] [MathSciNet] [Google Scholar]
  32. R. Glowinski and O. Pironneau, Numerical methods for the first biharmonic equation and for the two-dimensional Stokes problem. SIAM Rev. 21 (1979) 167–212. [CrossRef] [MathSciNet] [Google Scholar]
  33. T. Gudi, N. Nataraj and A.K. Pani, Mixed discontinuous Galerkin finite element method for the biharmonic equation. J. Sci. Comput. 37 (2008) 139–161. [Google Scholar]
  34. T. Gyulov and G. Morosanu, On a class of boundary value problems involving the p-biharmonic operator. J. Math. Anal. Appl. 367 (2010) 43–57. [CrossRef] [MathSciNet] [Google Scholar]
  35. N. Katzourakis and T. Pryer, On the numerical approximation of p-biharmonic and ∞-biharmonic functions. Numer. Methods Part. Differ. Eq. 35 (2019) 155–180. [CrossRef] [Google Scholar]
  36. U. Langer, On the numerical solution of the first biharmonic boundary value problem. Numer. Math. 50 (1986) 291–310. [CrossRef] [Google Scholar]
  37. A.M. Micheletti, A. Pistoia and C. Saccon, Three solutions of a fourth order elliptic problem via variational theorems of mixed type. Appl. Anal. 75 (2000) 43–59. [CrossRef] [MathSciNet] [Google Scholar]
  38. I. Mozolevski and E. Süli, A priori error analysis for the hp-version of the discontinuous Galerkin finite element method for the biharmonic equation. Comput. Methods Appl. Math. 3 (2003) 1–12. [Google Scholar]
  39. I. Mozolevski, E. Süli and P.R. Bösing, hp-Version a priori error analysis of interior penalty discontinuous Galerkin finite element approximations to the biharmonic equation. J. Sci. Comput. 30 (2007) 465–491. [Google Scholar]
  40. T. Pryer, Discontinuous Galerkin methods for the p-biharmonic equation from a discrete variational perspective. Electron. Trans. Numer. Anal. 41 (2014) 328–349. [MathSciNet] [Google Scholar]
  41. N. Rajashekar, S. Chaudhary and V.V.K. Srinivas Kumar, Approximation of p-biharmonic problem using WEB-spline based mesh-free method. Int. J. Nonlinear Sci. Numer. Simul. 20 (2019) 703–712. [CrossRef] [MathSciNet] [Google Scholar]
  42. S.I. Repin, A posteriori error estimation for variational problems with uniformly convex functionals. Math. Comp. 69 (2000) 481–500. [Google Scholar]
  43. S.I. Repin, A Posteriori Estimates for Partial Differential Equations. Vol. 4, Radon Series on Computational and Applied Mathematics. De Gruyter, Berlin (2008). [CrossRef] [Google Scholar]
  44. H.L. Royden and P. Fitzpatrick, Real Analysis. 4th edition. Phi Learning Pvt Ltd., New Delhi (2011). [Google Scholar]
  45. R. Scholz, A mixed method for 4th order problems using linear finite elements. RAIRO Anal. Numér./Numer. Anal. 12 (1978) 85–90. [CrossRef] [EDP Sciences] [MathSciNet] [Google Scholar]
  46. E. Süli and I. Mozolevski, hp-version interior penalty DGFEMs for the biharmonic equation. Comput. Methods Appl. Mech. Eng. 196 (2007) 1851–1863. [Google Scholar]
  47. L. Tartar, Introduction to Sobolev Spaces and Interpolation Theory. Springer, Berlin–Heidelberg–New York (2007). [Google Scholar]
  48. R. Verfürth, A Posteriori Error Estimation Techniques for Finite Element Methods. Oxford University Press, Oxford (2013). [CrossRef] [Google Scholar]
  49. W. Wang and P. Zhao, Nonuniformly nonlinear elliptic equations of p-biharmonic type. J. Math. Anal. Appl. 348 (2008) 730–738. [CrossRef] [MathSciNet] [Google Scholar]
  50. J.H. Zhang and S.J. Li, Multiple nontrivial solutions for some fourth-order semilinear elliptic problems. Nonlinear Anal. 60 (2005) 221–230. [CrossRef] [MathSciNet] [Google Scholar]

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.

Recommended for you