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All issues Volume 55 / No 5 (September-October 2021) ESAIM: M2AN, 55 5 (2021) 2075-2099 References
Open Access
Issue
ESAIM: M2AN
Volume 55, Number 5, September-October 2021
Page(s) 2075 - 2099
DOI https://doi.org/10.1051/m2an/2021034
Published online 01 October 2021
  1. D.N. Arnold, R.S. Falk and R. Winther, Finite element exterior calculus, homological techniques, and applications. Acta Numerica 15 (2006) 1–155. [Google Scholar]
  2. D.N. Arnold, R.S. Falk and R. Winther, Geometric decompositions and local bases for spaces of finite element differential forms. Comput. Methods Appl. Mech. Eng. 198 (2009) 1660–1672. [Google Scholar]
  3. D.N. Arnold, R.S. Falk and R. Winther, Finite element exterior calculus: from Hodge theory to numerical stability. Bull. Am. Math. Soc. 47 (2010) 281–354. [Google Scholar]
  4. R.E. Bank and H. Yserentant, A note on interpolation, best approximation, and the saturation property. Numerische Mathematik 131 (2015) 199–203. [Google Scholar]
  5. F. Bonizzoni, A. Buffa and F. Nobile, Moment equations for the mixed formulation of the Hodge Laplacian with stochastic loading term. IMA J. Numer. Anal. 34 (2014) 1328–1360. [Google Scholar]
  6. F. Camacho and A. Demlow, # and pointwise a posteriori error estimates for FEM for elliptic PDEs on surfaces. IMA J. Numer. Anal. 35 (2015) 1199–1227. [CrossRef] [Google Scholar]
  7. T. Chaumont-Frelet, M. Vohralk, Equivalence of local-best and global-best approximations in H(curl). HAL Preprint: hal-02736200 (2020). [Google Scholar]
  8. S.H. Christiansen, Stability of Hodge decompositions in finite element spaces of differential forms in arbitrary dimension. Numerische Mathematik 107 (2007) 87–106. [Google Scholar]
  9. S.H. Christiansen and M.W. Licht, Poincaré-Friedrichs inequalities of complexes of discrete distributional differential forms. BIT Numer. Math. 60 (2020) 345–371. [Google Scholar]
  10. S.H. Christiansen, H.Z. Munthe-Kaas and B. Owren, Topics in structure-preserving discretization. Acta Numerica 20 (2011) 1. [Google Scholar]
  11. S.H. Christiansen and R. Winther, Smoothed projections in finite element exterior calculus. Math. Comput. 77 (2008) 813–829. [CrossRef] [Google Scholar]
  12. P. Clément, Approximation by finite element functions using local regularization, Revue française d’automatique, informatique, recherche opérationnelle. Anal. numérique 9 (1975) 77–84. [Google Scholar]
  13. A. Demlow and A. Hirani, A posteriori error estimates for finite element exterior calculus: The de Rham complex. Found. Comput. Math. 12 (2014) 1–35. [Google Scholar]
  14. E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces. Bull. des sciences mathématiques 136 (2012) 521–573. [Google Scholar]
  15. T. Dupont and R. Scott, Polynomial approximation of functions in Sobolev spaces. Math. Comput. 34 (1980) 441–463. [Google Scholar]
  16. A. Ern, T. Gudi, I. Smears and M. Vohralk, Equivalence of local- and global-best approximations, a simple stable local commuting projector, and optimal hp approximation estimates in H(div). ArXiv e-prints [arXiv:1908.08158] (2019). [Google Scholar]
  17. A. Ern and J.-L. Guermond, Mollification in strongly Lipschitz domains with application to continuous and discrete de Rham complexes. Comput. Methods Appl. Math. 16 (2016) 51–75. [Google Scholar]
  18. A. Ern and J.-L. Guermond, Finite element quasi-interpolation and best approximation. ESAIM: M2AN 51 (2017) 1367–1385. [EDP Sciences] [Google Scholar]
  19. R. Falk and R. Winther, Local bounded cochain projections. Math. Comput. 83 (2014) 2631–2656. [Google Scholar]
  20. P. Fernandes and G. Gilardi, Magnetostatic and electrostatic problems in inhomogeneous anisotropic media with irregular boundary and mixed boundary conditions. Math. Models and Methods Appl. Sci. 7 (1997) 957–991. [Google Scholar]
  21. M. Fortin, An analysis of the convergence of mixed finite element methods. RAIRO. Anal. numérique 11 (1977) 341–354. [Google Scholar]
  22. V. Gol’dshtein, I. Mitrea and M. Mitrea, Hodge decompositions with mixed boundary conditions and applications to partial differential equations on Lipschitz manifolds. J. Math. Sci. 172 (2011) 347–400. [Google Scholar]
  23. V.M. Gol’dshtein, V.I. Kuz’minov and I.A. Shvedov, Differential forms on Lipschitz manifolds. Siberian Math. J. 23 (1982) 151–161. [Google Scholar]
  24. J. Gopalakrishnan and W. Qiu, Partial expansion of a Lipschitz domain and some applications. Fron. Math. China 7 (2011) 1–24. [Google Scholar]
  25. R. Hiptmair, Finite elements in computational electromagnetism. Acta Numerica 11 (2002) 237–339. [Google Scholar]
  26. T. Iwaniec, C. Scott and B. Stroffolini, Nonlinear Hodge theory on manifolds with boundary. Annali di Matematica pura ed applicata 177 (1999) 37–115. [Google Scholar]
  27. T. Jakab, I. Mitrea and M. Mitrea, On the regularity of differential forms satisfying mixed boundary conditions in a class of Lipschitz domains. Ind. Univ. Math. J. 58 (2009) 2043–2072. [Google Scholar]
  28. F. Jochmann, A compactness result for vector fields with divergence and curl in Lq (Ω) involving mixed boundary conditions. Appl. Anal. 66 (1997) 189–203. [CrossRef] [Google Scholar]
  29. F. Jochmann, Regularity of weak solutions of Maxwell’s equations with mixed boundary-conditions. Math. Methods Appl. Sci. 22 (1999) 1255–1274. [Google Scholar]
  30. M. Licht, Smoothed projections and mixed boundary conditions. Math. Comput. 88 (2019) 607–635. [Google Scholar]
  31. M. Licht, Smoothed projections over weakly Lipschitz domains. Math. Comput. 88 (2019) 179–210. [Google Scholar]
  32. M.W. Licht, Complexes of discrete distributional differential forms and their homology theory. Found. Comput. Math. 17 (2017) 1085–1122. [Google Scholar]
  33. M.W. Licht, On the a priori and a posteriori error analysis in finite element exterior calculus, Ph.D. thesis. Dissertation, Department of Mathematics, University of Oslo, Norway (2017). [Google Scholar]
  34. D. Mitrea, M. Mitrea and M. Shaw, Traces of differential forms on Lipschitz domains, the boundary de Rham complex, and Hodge decompositions. J. Funct. Anal. 190 (2002) 339–417. [Google Scholar]
  35. P. Oswald, On a BPX-preconditioner for P1 elements. Comput. 51 (1993) 125–133. [Google Scholar]
  36. J. Schöberl, A posteriori error estimates for Maxwell equations. Math. Comput. 77 (2008) 633–649. [Google Scholar]
  37. C. Scott, Lp theory of differential forms on manifolds. Trans. Am. Math. Soc. 347 (1995) 2075–2096. [Google Scholar]
  38. L.R. Scott and S. Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput. 54 (1990) 483–493. [Google Scholar]
  39. L. Slobodeckij, Generalized Sobolev spaces and their applications to boundary value problems of partial differential equations. Gos. Ped. Inst. Ucep. Zap 197 (1958) 54–112. [Google Scholar]
  40. A. Veeser, Approximating gradients with continuous piecewise polynomial functions. Found. Comput. Math. 16 (2016) 723–750. [CrossRef] [MathSciNet] [Google Scholar]

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