Open Access
Issue
ESAIM: M2AN
Volume 56, Number 2, March-April 2022
Page(s) 385 - 406
DOI https://doi.org/10.1051/m2an/2022005
Published online 14 February 2022
  1. R.A. Adams, Sobolev spaces, Academic Press (1975). [Google Scholar]
  2. R. Araya, E. Behrens and R. Rodríguez, A posteriori error estimates for elliptic problems with Dirac delta source terms. Numer. Math. 105 (2006) 193–216. [CrossRef] [MathSciNet] [Google Scholar]
  3. R. Araya, E. Behrens and R. Rodríguez, An adaptive stabilized finite element scheme for a water quality model. Comput. Methods Appl. Mech. Engry. 196 (2007) 2800–2812. [CrossRef] [Google Scholar]
  4. R. Araya, M. Solano and P. Vega, A posteriori error analysis of an HDG method for the Oseen problem. Appl. Numer. Math. 146 (2019) 291–308. [CrossRef] [MathSciNet] [Google Scholar]
  5. R. Araya, M. Solano and P. Vega, Analysis of an adaptive HDG method for the Brinkman problem. IMA J. Numer. Anal. 39 (2019) 1502–1528. [Google Scholar]
  6. T. Apel, O. Benedix, D. Sirch and B. Vexler, A priori mesh grading for an elliptic problem with Dirac right-hand side. SIAM J. Numer. Anal. 49 (2011) 992–1005. [Google Scholar]
  7. J.P. Agnelli, E.M. Garau and P. Morin, A posteriori error estimates for elliptic problems with Dirac measure terms in weighted spaces. ESAIM: M2AN 48 (2014) 1557–1581. [CrossRef] [EDP Sciences] [Google Scholar]
  8. I. Babuška, Error bounds for the finite element method. Numer. Math. 16 (1971) 322–333. [CrossRef] [Google Scholar]
  9. S.C. Brenner and L.R. Scott, The mathematical theory of finite element methods. Springer-Verlag, New York (2008). [CrossRef] [Google Scholar]
  10. P.G. Ciarlet, The finite element methods for elliptic problems. Stud. Math. Appl. vol. 4, North-Holland, Amsterdam (1978). [Google Scholar]
  11. E. Casas, L2 estimates for the finite element method for the Dirchlet problem with singular data. Numer. Math. 47 (1985) 627–632. [CrossRef] [MathSciNet] [Google Scholar]
  12. E. Casas, Control of an elliptic problem with pointwise state constraints. SIAM J. Control Optim. 24 (1986) 1309–1318. [CrossRef] [MathSciNet] [Google Scholar]
  13. B. Cockburn and W. Zhang, A posteriori error estimates for HDG methods. J. Sci. Comput. 51 (2012) 582–607. [CrossRef] [MathSciNet] [Google Scholar]
  14. B. Cockburn and W. Zhang, A posteriori error analysis for hybridizable discontinuous Galerkin methods for second order elliptic problems. SIAM J. Numer. Anal. 51 (2013) 676–693. [CrossRef] [MathSciNet] [Google Scholar]
  15. B. Cockburn, J. Gopalakrishnan and R. Lazarov, Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems. SIAM J. Numer. Anal. 47 (2009) 1319–1365. [Google Scholar]
  16. B. Cockburn, J. Gopalakrishnan, N.C. Nguyen, J. Peraire and F.-J. Sayas, Analysis of HDG methods for Stokes flow. Math. Comput. 80 (2011) 723–760. [CrossRef] [Google Scholar]
  17. H. Chen, J. Li and W. Qiu, A posteriori error estimates for HDG method for convection-diffusion equations. IMA J. Numer. Anal. 36 (2016) 437–462. [MathSciNet] [Google Scholar]
  18. G. Chen, W. Hu, J. Shen, J.R. Singler, Y. Zhang and X. Zheng, An HDG method for distributed control of convection diffusion PDEs. J. Comput. Appl. Math. 343 (2018) 643–661. [Google Scholar]
  19. K. Eriksson, Improve accuarcy by adapted mesh-refinements in the finite element method. Math. Comput. 44 (1985) 321–343. [CrossRef] [Google Scholar]
  20. F. Fuica, F. Lepe, E. Otárola and D. Quero, A posteriori error estimates in W1,p × Lp spaces for the Stokes system with Dirac measures. Preprint arXiv:1912.08325 (2019). [Google Scholar]
  21. G. Fu, W. Qiu and W. Zhang, An analysis of HDG methods for convection-dominated diffusion problems. ESAIM: M2AN 49 (2015) 225–256. [CrossRef] [EDP Sciences] [Google Scholar]
  22. W. Gong, G. Wang and N. Yan, Approximation of elliptic optimal control problems acting on a lower dimensional manifold. SIAM J. Control Optim. 52 (2014) 2008–2035. [CrossRef] [MathSciNet] [Google Scholar]
  23. W. Gong, W. Hu, M. Mateos, J.R. Singler, X. Zheng and Y. Zhang, A new HDG method for Dirichlet boundary control of convection diffusion PDEs II: low regularity. SIAM J. Numer. Anal. 56 (2018) 2262–2287. [CrossRef] [MathSciNet] [Google Scholar]
  24. P. Grisvard, Elliptic problems for non-smooth domains. Pitman, Boston (1985). [Google Scholar]
  25. F.D. Gaspoz, P. Morin and A. Veeser, A posteriori error estimates with point sources in fractional Sobolev spaces. Numer. Methods Part. Diff. Equ. 33 (2017) 1018–1042. [CrossRef] [Google Scholar]
  26. P. Houston and T.P. Wihler, Discontinuous Galerkin methods for problems with Dirac delta source. ESAIM: M2AN 46 (2012) 1467–1483. [CrossRef] [EDP Sciences] [Google Scholar]
  27. O.A. Karakashian and F. Pascal, A posteriori error estimates for a discontinuous Galerkin approximation of second-order elliptic problems. SIAM J. Numer. Anal. 41 (2003) 2374–2399. [Google Scholar]
  28. H. Leng, Adaptive HDG methods for the steady-state incompressible Navier-Stokes equations. J. Sci. Comput. 87 (2021) 37. [CrossRef] [Google Scholar]
  29. H. Leng and H. Chen, Adaptive HDG methods for the Brinkman equations with application to optimal control. J. Sci. Comput. 87 (2021) 46. [CrossRef] [Google Scholar]
  30. I. Oikawa, HDG methods for second order elliptic problems. RIMS Kokyuroku 2037 (2017) 61–74. [Google Scholar]
  31. M. Petzoldt, A posteriori error estimators for elliptic equations with discontinuous coefficients. Adv. Comput. Math. 16 (2002) 47–75. [CrossRef] [MathSciNet] [Google Scholar]
  32. W. Qiu and K. Shi, A superconvergent HDG method for the incompressible Navier-Stokes equations on general polyhedral meshes. IMA J. Numer. Anal. 36 (2016) 1943–1967. [CrossRef] [MathSciNet] [Google Scholar]
  33. R. Scott, Finite element convergence for singular data. Numer. Math. 21 (1973) 317–327. [Google Scholar]
  34. A.H. Schatz and L.B. Wahlbin, Interior maxumum norm estimates for finite element methods. Math. Comput. 31 (1977) 414–442. [CrossRef] [Google Scholar]
  35. R. Verfürth, A posteriori error estimators for convection-diffusion equations. Numer. Math. 80 (1998) 641–663. [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