Open Access
Issue
ESAIM: M2AN
Volume 58, Number 4, July-August 2024
Page(s) 1347 - 1383
DOI https://doi.org/10.1051/m2an/2024037
Published online 30 July 2024
  1. R. Adam, Sobolev Spaces. Academic Press, New York (1975). [Google Scholar]
  2. K. Allali, A priori and a posteriori error estimates for Boussinesq equations. Int. J. Numer. Anal. Model. 2 (2005) 179–196. [MathSciNet] [Google Scholar]
  3. M. Ainsworth and J.T. Oden, A posteriori error estimation in finite element analysis, in Pure and Applied Mathematics. Wiley-Insterscience, New York (2000). [Google Scholar]
  4. 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]
  5. A. Allendes, G.R. Barrenechea and C. Naranjo, A divergence-free low-order stabilized finite element method for a generalized steady state Boussinesq problem. Comput. Methods Appl. Mech. Eng. 340 (2018) 90–120. [CrossRef] [Google Scholar]
  6. A. Allendes, C. Naranjo and E. Otárola, Stabilized finite element approximations for a generalized Boussinesq problem: a posteriori error analysis. Comput. Methods Appl. Mech. Eng. 361 (2020) 112703. [CrossRef] [Google Scholar]
  7. A. Allendes, E. Otárola and A.J. Salgado, The stationary Boussinesq problem under singular forcing. Math. Models Methods Appl. Sci. 31 (2021) 789–827. [CrossRef] [MathSciNet] [Google Scholar]
  8. A. Allendes, F. Fuica and E. Otárola, Error estimates for a pointwise tracking optimal control problem of a semilinear elliptic equation. SIAM J. Control Optim. 60 (2022) 1763–1790. [CrossRef] [MathSciNet] [Google Scholar]
  9. A. Allendes, G. Campaña and E. Otárola, Numerical discretization of a Darcy-Forchheimer problem coupled with a singular heat equation. SIAM J. Sci. Comput. 45 (2023) A2755–A2780. [CrossRef] [Google Scholar]
  10. A. Allendes, G. Campaña, F. Fuica and E. Otárola, Darcy’s problem coupled with the heat equation under singular forcing: analysis and discretization. IMA J. Numer. Anal. (2024) drad094. [CrossRef] [Google Scholar]
  11. J.A. Almonacid and G.N. Gatica, A fully-mixed finite element method for the n-dimensional Boussinesq problem with temperature-dependent parameters. Comput. Methods Appl. Math. 20 (2020) 187–213. [CrossRef] [MathSciNet] [Google Scholar]
  12. J.A. Almonacid, G.N. Gatica and R. Oyarzúa, A mixed-primal finite element method for the Boussinesq problem with temperature-dependent viscosity. Calcolo 55 (2018) 36. [CrossRef] [Google Scholar]
  13. J.A. Almonacid, G.N. Gatica and R. Oyarzúa, A posteriori error analysis of a mixed-primal finite element method for the Boussinesq problem with temperature-dependent viscosity. J. Sci. Comput. 78 (2019) 887–917. [CrossRef] [MathSciNet] [Google Scholar]
  14. A. Alonso Rodríguez, J. Camaño, R. Rodríguez and A. Valli, A posteriori error estimates for the problem of electrostatics with a dipole source. Comput. Math. Appl. 68 (2014) 464–485. [CrossRef] [MathSciNet] [Google Scholar]
  15. C. Bernardi, B. Métivet and B. Pernaud-Thomas, Couplage des équations de Navier–Stokes et de la chaleur: le modèle et son approximation par éléments finis. RAIRO Modél. Math. Anal. Numér. 29 (1995) 871–921. [MathSciNet] [Google Scholar]
  16. J. Boland and W. Layton, An analysis of the finite element method for natural convection problems. Numer. Methods Part. Differ. Equ. 6 (1990) 115–126. [CrossRef] [Google Scholar]
  17. S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods. Springer-Verlag, New York (2008). [Google Scholar]
  18. E. Casas, Control of an elliptic problem with pointwise state constaints. SIAM J. Control Optim. 24 (1986) 1309–1318. [CrossRef] [MathSciNet] [Google Scholar]
  19. A. Cesmelioglu, B. Cockburn and W. Qiu, Analysis of a hybridizable discontinuous Galerkin method for the steady-state incompressible Navier–Stokes equations. Math. Comput. 86 (2017) 1643–1670. [Google Scholar]
  20. 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. [CrossRef] [MathSciNet] [Google Scholar]
  21. D.A. Di Pietro and A. Ern, Discrete functional analysis tools for discontinuous Galerkin methods with application to the incompressible Navier–Stokes equations. Math. Comput. 79 (2010) 1303–1330. [CrossRef] [Google Scholar]
  22. D.A. Di Pietro and A. Ern, Mathematical Aspects of Discontinuous Galerkin Methods. Springer, Heidelberg (2012). [Google Scholar]
  23. A. Ern and J.-L. Guermond, Theory and Practice of Finite Elements. Springer-Verlag, New York (2004). [CrossRef] [Google Scholar]
  24. M. Farhloul, S. Nicaise and L. Paquet, A mixed formulation of Boussinesq equations: analysis of nonsingular solutions. Math. Comput. 69 (2000) 965–986. [CrossRef] [Google Scholar]
  25. 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]
  26. F. Fuica, F. Lepe, E. Otárola and D. Quero, An optimal control problem for the Navier–Stokes equations with point sources. J. Optim. Theory Appl. 196 (2023) 590–616. [CrossRef] [MathSciNet] [Google Scholar]
  27. G. Fulford and J. Blake, Muco-ciliary transport in the lung. J. Theor. Biol. 121 (1986) 381–402. [CrossRef] [Google Scholar]
  28. V. Girault and P.-A. Raviart, Finite Element Methods for Navier–Stokes Equations, Theory and Algorithms. Springer-Verlag, Berlin Heidelberg (1986). [CrossRef] [Google Scholar]
  29. J. Guzmán, Pointwise error estimates for discontinuous Galerkin methods with lifting operators for elliptic problems. Math. Comput. 75 (2006) 1067–1085. [CrossRef] [Google Scholar]
  30. P. Hansbo and M.G. Larson, Discontinuous Galerkin methods for incompressible and nearly incompressible elasticity by Nitsche’s method. Comput. Methods Appl. Mech. Eng. 191 (2002) 1895–1908. [CrossRef] [Google Scholar]
  31. P. Houston and T.P. Wihler, Discontinuous Galerkin methods for problem with Dirac delta source. ESAIM: M2AN 46 (2012) 1467–1483. [CrossRef] [EDP Sciences] [Google Scholar]
  32. D. Jerison and C.E. Kenig, The inhomogeneous Dirichlet problem in Lipschitz domain. J. Funct. Anal. 130 (1995) 161–219. [CrossRef] [MathSciNet] [Google Scholar]
  33. O.A. Karakashian and W.N. Jureidini, A nonconforming finite element method for the stationary Navier–Stokes equations. SIAM J. Numer. Anal. 35 (1998) 93–120. [CrossRef] [MathSciNet] [Google Scholar]
  34. 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. [CrossRef] [MathSciNet] [Google Scholar]
  35. K.L.A. Kirk and S. Rhebergen, Analysis of a pressure-robust hybridized discontinuous Galerkin method for the stationary Navier–Stokes equations. J. Sci. Comput. 81 (2019) 881–897. [CrossRef] [MathSciNet] [Google Scholar]
  36. H. Leng, Analysis of an HDG method for the Navier–Stokes equations with Dirac measures. ESAIM: M2AN 57 (2023) 271–297. [CrossRef] [EDP Sciences] [MathSciNet] [Google Scholar]
  37. S.A. Lorca and J.L. Boldrini, Stationary solutions for generalized Boussinesq models. J. Differ. Equ. 124 (1996) 389–406. [CrossRef] [Google Scholar]
  38. M.T. Manzari, An explicit finite element algorithm for convection heat transfer problems. Int. J. Numer. Methods Heat Fluid Flow 9 (1999) 860–877. [CrossRef] [Google Scholar]
  39. N. Nguyen, J. Peraire and B. Cockburn, An implict high-order hybridizable discontinuous Galerkin method for linear convection diffusion equations. J. Comput. Phys. 228 (2009) 3232–3254. [CrossRef] [MathSciNet] [Google Scholar]
  40. N. Nguyen, J. Peraire and B. Cockburn, An implict high-order hybridizable discontinuous Galerkin method for the incompressible Navier–Stokes equations. J Comput. Phys. 230 (2011) 1147–1170. [CrossRef] [MathSciNet] [Google Scholar]
  41. E. Otárola, Semilinear optimal control with Dirac measures. IMA J. Numer. Anal. (2023) drad091. [CrossRef] [Google Scholar]
  42. R. Oyarzúa and M. Serón, A divergence-conforming DG-mixed finite element method for the stationary Boussinesq problem. J. Sci. Comput. 85 (2020) 14. [CrossRef] [Google Scholar]
  43. R. Oyarzúa and P. Zúñiga, Analysis of a conforming finite element method for the Boussinesq problem with temperature-denpendent parameters. J. Comput. Appl. Math. 323 (2017) 71–94. [CrossRef] [MathSciNet] [Google Scholar]
  44. R. Oyarzúa, T. Qin and D. Schötzau, An exactly divergence-free finite element method for a generalized Boussinesq problem. IMA J. Numer. Anal. 34 (2014) 1104–1135. [CrossRef] [MathSciNet] [Google Scholar]
  45. C.E. Pérez, J.-M. Thomas, S. Blancher and R. Creff, The steady Navier–Stokes/energy system with temperature-dependent viscosity. I. Analysis of the continuous problem. Int. J. Numer. Methods Fluids 56 (2008) 63–89. [CrossRef] [Google Scholar]
  46. C.E. Pérez, J.-M. Thomas, S. Blancher and R. Creff, The steady Navier–Stokes/energy system with temperature-dependent viscosity. II. The discrete problem and numerical experiments. Int. J. Numer. Methods Fluids 56 (2008) 91–114. [CrossRef] [Google Scholar]
  47. S. Rhebergen and G.N. Wells, A hybridizable discontinuous Galerkin method for the Navier–Stokes equations with pointwise divergence-free velocity field. J. Sci. Comput. 76 (2018) 1484–1501. [CrossRef] [MathSciNet] [Google Scholar]
  48. S. Rhebergen and G.N. Wells, Analysis of a hybridized/interface stabilized finite element method for the Stokes equations. SIAM J. Numer. Anal. 55 (2017) 1982–2003. [CrossRef] [MathSciNet] [Google Scholar]
  49. S. Rhebergen and G.N. Wells, Preconditioning of a hybridized discontinuous Galerkin finite element method for the Stokes equations. J. Sci. Comput. 77 (2018) 1936–1952. [CrossRef] [MathSciNet] [Google Scholar]
  50. Z. Shen, Bounds of Riesz transforms on Lp spaces for second order elliptic operators. Ann. Inst. Fourier 55 (2005) 173–197. [CrossRef] [MathSciNet] [Google Scholar]
  51. M.P. Ueckermann and P.F.J. Lermusiaux, Hybridizable discontinuous Galerkin projection methods for Navier–Stokes and Boussinesq equations. J. Comput. Phys. 306 (2016) 390–421. [CrossRef] [MathSciNet] [Google Scholar]
  52. R. Verfürth, A posteriori error estimators for convection-diffusion equations. Numer. Math. 80 (1998) 641–663. [CrossRef] [MathSciNet] [Google Scholar]
  53. R. Verfürth, A Posteriori Error Estimation Techniques for Finite Element Methods, Numerical Mathematics on Scientific Computation. Oxford University Press, Oxford (2013). [CrossRef] [Google Scholar]
  54. G.N. Wells, Analysis of an interface stabilized finite element method: the advection-diffusion-reaction equation. SIAM J. Numer. Anal. 49 (2011) 87–109. [CrossRef] [MathSciNet] [Google Scholar]

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