Open Access
Issue
ESAIM: M2AN
Volume 58, Number 2, March-April 2024
Page(s) 457 - 488
DOI https://doi.org/10.1051/m2an/2024006
Published online 09 April 2024
  1. R. Adams and J. Fournier, Cone conditions and properties of Sobolev spaces. J. Math. Anal. App. 61 (1977) 713–734. [Google Scholar]
  2. N. Ahmed and G. Matthies, Higher-order discontinuous Galerkin time discretizations for the evolutionary Navier–Stokes equations. IMA J. Numer. Anal. 41 (2021) 3113–3144. [Google Scholar]
  3. H. Amann, Linear parabolic problems involving measures. Real Academia de Ciencias Exactas, Fisicas y Naturales. Revista. Serie A, Matematicas 95 (2001) 85–119. [Google Scholar]
  4. D. Arndt, H. Dallmann and G. Lube, Local projection FEM stabilization for the time-dependent incompressible Navier–Stokes problem. Numer. Methods Part. Differ. Equ. 31 (2015) 1224–1250. [Google Scholar]
  5. D. Arndt, H. Dallmann and G. Lube, Quasi-optimal error estimates for the incompressible Navier–Stokes problem discretized by finite element methods and pressure-correction projection with velocity stabilization. Preprint arXiv:1609.00807 (2016). [Google Scholar]
  6. A. Ashyralyev and P.E. Sobolevskii, Well-Posedness of Parabolic Difference Equations. Birkh¨auser Basel (1994). [Google Scholar]
  7. N. Behringer, B. Vexler and D. Leykekhman, Fully discrete best-approximation-type estimates in L(I; L2(Ω)d) for finite element discretizations of the transient Stokes equations. IMA J. Numer. Anal. 43 (2022) 852–880. [Google Scholar]
  8. J. Bergh and J. Löfström, Interpolation Spaces: An Introduction. Vol. 223 of Grundlehren der mathematischen Wissenschaften. Springer Berlin Heidelberg (1976). [Google Scholar]
  9. C. Bernardi and G. Raugel, A conforming finite element method for the time-dependent Navier–Stokes equations. SIAM J. Numer. Anal. 22 (1985) 455–473. [Google Scholar]
  10. I. Bihari, A generalization of a lemma of Bellman and its application to uniqueness problems of differential equations. Acta Math. Acad. Sci. Hung. 7 (1956) 81–94. [Google Scholar]
  11. H. Brezis and P. Mironescu, Gagliardo–Nirenberg inequalities and non-inequalities: the full story. Annales de l’Institut Henri Poincaré C, Analyse non linéaire 35 (2018) 1355–1376. [Google Scholar]
  12. M. Brokate and G. Kersting, Measure and Integral. Springer International Publishing (2015). [Google Scholar]
  13. E. Casas and K. Chrysafinos, A discontinuous Galerkin time-stepping scheme for the velocity tracking problem. SIAM J. Numer. Anal. 50 (2012) 2281–2306. [Google Scholar]
  14. E. Casas and K. Chrysafinos, Analysis of the velocity tracking control problem for the 3D evolutionary Navier–Stokes equations. SIAM J. Control Optim. 54 (2016) 99–128. [Google Scholar]
  15. E. Casas and K. Kunisch, Well-posedness of evolutionary Navier–Stokes equations with forces of low regularity on two-dimensional domains. ESAIM: Control Optim. Calculus Variations 27 (2021). [Google Scholar]
  16. K. Chrysafinos and N. Walkington, Discontinuous Galerkin approximations of the Stokes and Navier–Stokes equations. Math. Comput. 79 (2010) 2135–2167. [Google Scholar]
  17. M. Dauge, Stationary Stokes and Navier–Stokes systems on two- or three-dimensional domains with corners. Part I. Linearized equations. SIAM J. Math. Anal. 20 (1989) 74–97. [Google Scholar]
  18. J. De Frutos, B. García-Archilla, V. John and J. Novo, Error analysis of non inf-sup stable discretizations of the time-dependent Navier–Stokes equations with local projection stabilization. IMA J. Numer. Anal. 39 (2019) 1747–1786. [Google Scholar]
  19. L. de Simon, Un’applicazione della teoria degli integrali singolari allo studio delle equazioni differenziali lineari astratte del primo ordine. Rendiconti del Seminario Matematico della Università di Padova 34 (1964) 205–223. [Google Scholar]
  20. K. Deckelnick and M. Hinze, Semidiscretization and error estimates for distributed control of the instationary Navier–Stokes equations. Numer. Math. 97 (2004) 297–320. [Google Scholar]
  21. F. Demengel and G. Demengel, Functional Spaces for the Theory of Elliptic Partial Differential Equations. Universitext, Springer London, (2012). [Google Scholar]
  22. K. Disser, A. ter Elst and J. Rehberg, Hölder estimates for parabolic operators on domains with rough boundary. Annali Scuola Normale Superiore di Pisa, Classe di Scienze XVII (2017) 65–79. [Google Scholar]
  23. E. Emmrich, Error of the two-step BDF for the incompressible Navier–Stokes problem. ESAIM: Math. Modell. Numer. Anal. 38 (2004) 757–764. [Google Scholar]
  24. K. Eriksson, C. Johnson and V. Thomée, Time discretization of parabolic problems by the discontinuous Galerkin method. ESAIM: Math. Modell. Numer. Anal. 19 (1985) 611–643. [Google Scholar]
  25. F. Gabel and P. Tolksdorf, The Stokes operator in two-dimensional bounded Lipschitz domains. J. Differ. Equ. 340 (2022) 227–272. [Google Scholar]
  26. G. Galdi, An Introduction to the Mathematical Theory of the Navier–Stokes Equations. Springer Monographs in Mathematics. Springer New York (2011). [Google Scholar]
  27. V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations. Vol. 5 of Springer Series in Computational Mathematics. Springer Berlin Heidelberg (1986). [Google Scholar]
  28. J.-L. Guermond and J.E. Pasciak, Stability of discrete Stokes operators in fractional Sobolev spaces. J. Math. Fluid Mech. 10 (2008) 588–610. [Google Scholar]
  29. A. Hansbo, Strong stability and non-smooth data error estimates for discretizations of linear parabolic problems. BIT Numer. Math. 42 (2002) 351–379. [Google Scholar]
  30. J.G. Heywood and R. Rannacher, Finite element approximation of the nonstationary Navier–Stokes problem. I. Regularity of solutions and second-order error estimates for spatial discretization. SIAM J. Numer. Anal. 19 (1982) 275–311. [Google Scholar]
  31. J.G. Heywood and R. Rannacher, Finite element approximation of the nonstationary Navier–Stokes problem, part II: stability of solutions and error estimates uniform in time. SIAM J. Numer. Anal. 23 (1986) 750–777. [Google Scholar]
  32. J.G. Heywood and R. Rannacher, Finite-element approximation of the nonstationary Navier–Stokes problem part IV: error analysis for second order time discretization. SIAM J. Numer. Anal. 27 (1990) 353–384. [Google Scholar]
  33. R. Kellogg and J. Osborn, A regularity result for the Stokes problem in a convex polygon. J. Funct. Anal. 21 (1976) 397–431. [Google Scholar]
  34. K. Kirk, T. Horváth and S. Rhebergen, Analysis of an exactly mass conserving space-time hybridized discontinuous Galerkin method for the time-dependent Navier–Stokes equations. Math. Comput. 92 (2022) 525–556. [Google Scholar]
  35. D. Leykekhman and B. Vexler, Discrete maximal parabolic regularity for Galerkin finite element methods. Numer. Math. 135 (2017) 923–952. [Google Scholar]
  36. D. Leykekhman and B. Vexler, L2(I; H1(Ω)) and L2(I; L2(Ω)) best approximation type error estimates for Galerkin solutions of transient Stokes problems. Calcolo 61 (2024) 7. [Google Scholar]
  37. D. Leykekhman, B. Vexler and D. Walter, Numerical analysis of sparse initial data identification for parabolic problems. ESAIM: Math. Modell. Numer. Anal. 54 (2020) 1139–1180. [Google Scholar]
  38. Z. Li, Some generalized Gronwall–Bellman type difference inequalities and applications. J. Math. Inequalities 15 (2021) 173–200. [Google Scholar]
  39. O. Lipovan, A retarded Gronwall-like inequality and its applications. J. Math. Anal. App. 252 (2000) 389–401. [Google Scholar]
  40. A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems. Modern Birkh¨auser classics. Birkh¨auser (2012). [Google Scholar]
  41. E. Magenes, Interpolational spaces and partial differential equations. Uspekhi Mat. Nauk 21 (1966) 169–218. [Google Scholar]
  42. D. Meidner, R. Rannacher and B. Vexler, A priori error estimates for finite element discretizations of parabolic optimization problems with pointwise state constraints in time. SIAM J. Control Optim. 49 (2011) 1961–1997. [Google Scholar]
  43. B. Pachpatte, Integral and Finite Difference Inequalities and Applications. Vol. 205 of North-Holland Mathematics Studies. Elsevier (2006). [Google Scholar]
  44. A.H. Schatz and L.B. Wahlbin, Interior maximum-norm estimates for finite element methods, part II. Math. Comput. 64 (1995) 907. [Google Scholar]
  45. F. Schieweck and L. Tobiska, An optimal order error estimate for an upwind discretization of the Navier–Stokes equations. Numer. Methods Part. Differ. Equ. 12 (1996) 407–421. [Google Scholar]
  46. J. Shen, On error estimates of projection methods for Navier–Stokes equations: first-order schemes. SIAM J. Numer. Anal. 29 (1992) 57–77. [Google Scholar]
  47. H. Sohr, The Navier–Stokes Equations. Springer Basel (2001). [Google Scholar]
  48. R. Temam, Navier–Stokes Equations: Theory and Numerical Analysis. North-Holland Publ. Co., Amsterdam (1977). [Google Scholar]
  49. V. Thomée, Galerkin Finite Element Methods for Parabolic Problems. Vol. 25 of Springer Series in Computational Mathematics, 2nd edition. Springer (2006). [Google Scholar]
  50. H. Triebel, Theory of Function Spaces II. Springer Basel (1992). [Google Scholar]

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