Open Access
Issue
ESAIM: M2AN
Volume 55, Number 5, September-October 2021
Page(s) 1963 - 2011
DOI https://doi.org/10.1051/m2an/2021043
Published online 29 September 2021
  1. W. Bangerth, R. Hartmann and G. Kanschat, deal.II – a general purpose object oriented finite element library. ACM Trans. Math. Softw. 33 (2007) 24-es. [CrossRef] [Google Scholar]
  2. S. Bartels, M. Jensen and R. Müller, Discontinuous Galerkin finite element convergence for incompressible miscible displacement problems of low regularity. SIAM J. Numer. Anal. 47 (2009) 3720–3743. [CrossRef] [Google Scholar]
  3. D. Boffi, Three dimensional finite element methods for the Stokes problem. SIAM J. Numer. Anal. 34 (1997) 664–670. [CrossRef] [MathSciNet] [Google Scholar]
  4. P. Boltenhagen, Y. Hu, E. Matthys and D. Pine, Observation of bulk phase separation and coexistence in a sheared micellar solution. Phys. Rev. Lett. 79 (1997) 2359–2362. [CrossRef] [Google Scholar]
  5. A. Bonito, V. Girault and E. Süli, Finite element approximation of a strain-limiting elastic model. IMA J. Numer. Anal. 40 (2020) 29–86. [CrossRef] [Google Scholar]
  6. S. Brenner, Poincaré-Friedrichs inequalities for piecewise H1 functions. SIAM J. Numer. Anal. 41 (2003) 306–324. [CrossRef] [MathSciNet] [Google Scholar]
  7. S. Brenner, Korn’s inequalities for piecewise H1 vector fields. Math. Comp. 73 (2004) 1067–1087. [Google Scholar]
  8. S. Brenner and L. Scott, The mathematical theory of finite element methods, 3rd edition. In: Vol. 15 of Texts in Applied Mathematics. Springer, New York (2008). [Google Scholar]
  9. P. Bridgman, The Physics of High Pressure. MacMillan (1931). [Google Scholar]
  10. 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]
  11. M. Bulček, P. Gwiazda, J. Málek, K. Rajagopal and A. Świerczewska-Gwiazda, On flows of fluids described by an implicit constitutive equation characterized by a maximal monotone graph, edited by J. Robinson, J. Rodrigo and W. Sadowski. In: Mathematical Aspects of Fluid Mechanics. London Mathematical Society Lecture Note Series. Cambridge University Press (2012) 23–51. [Google Scholar]
  12. M. Bulček, J. Málek, K. Rajagopal and E. Süli, On elastic solids with limiting small strain: modelling and analysis. EMS Surv. Math. Sci. 1 (2014) 283–332. [Google Scholar]
  13. J. Burgers, Mechanical considerations-Model systems-Phenomenological theories of relaxation and of viscosity. First report on viscosity and plasticity. Prepared by the Committee for the Study of Viscosity, 2nd edition. Tech. Report, Academy of Sciences at Amsterdam (1939). [Google Scholar]
  14. P. Ciarlet, Basic error estimates for elliptic problems. In: Vol. II of Handbook of Numerical Analysis. North-Holland, Amsterdam (1991) 17–351. [Google Scholar]
  15. M. Crouzeix and P.-A. Raviart, Conforming and nonconforming finite element methods for solving the stationary Stokes equations. RAIRO 7 (1973) 33–75. [Google Scholar]
  16. D. DiPietro 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] [MathSciNet] [Google Scholar]
  17. L. Evans, Partial differential equations. In: Vol. 19 of Graduate Studies in Mathematics. American Mathematical Society (2010). [Google Scholar]
  18. R. Falk, Nonconforming finite element methods for the equations of linear elasticity. Math. Comput. 57 (1991) 529–550. [CrossRef] [MathSciNet] [Google Scholar]
  19. V. Girault and P.-A. Raviart, Finite element methods for Navier–Stokes equations: theory and algorithms, In: Vol. 5 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin (1986). [Google Scholar]
  20. V. Girault, J. Li and B. Rivière, Strong convergence of discrete DG solutions to the heat equation. J. Numer. Math. 24 (2016) 235–252. [Google Scholar]
  21. T. Hu, P. Boltenhagen, E. Matthys and D. Pine, Shear thickening in low-concentration solutions of wormlike micelles. II. Slip, fracture, and stability of the shear-induced phase. J. Rheol. 42 (1998) 1209–1226. [CrossRef] [Google Scholar]
  22. A. Johnen, J.-C. Weill and J.-F. Remacle, Robust and efficient validation of the linear hexahedral element. Proc. Eng. 203 (2017) 271–283. [CrossRef] [Google Scholar]
  23. P. Knabner, S. Korotov and G. Summ, Conditions for the invertibility of the isoparametric mapping for hexahedral finite elements. Finite. Elem. Anal. Des. 40 (2003) 159–172. [CrossRef] [Google Scholar]
  24. A. Lasis and E. Süli, Poincaré-type inequalities for broken Sobolev spaces. Internal Report 03/10. Oxford University Computing Laboratory, Numerical Analysis Group, Oxford, England, OX1 3QD (2003). [Google Scholar]
  25. P.-L. Lions and B. Mercier, Splitting algorithms for the sum of two nonlinear operators. SIAM J. Numer. Anal. 16 (1979) 964–979. [CrossRef] [MathSciNet] [Google Scholar]
  26. D. Lopez-Diaz, E. Sarmiento-Gomez, C. Garza and R. Castillo, A rheological study in the dilute regime of the worm-micelle fluid made of zwitterionic surfactant (TDPS), anionic surfactant (SDS), and brine. J. Colloid Interface Sci. 348 (2010) 152–158. [CrossRef] [PubMed] [Google Scholar]
  27. J. Málek and K. Rajagopal, Mathematical properties of the equations governing the flow of fluids with pressure and shear rate dependent viscosities, edited by S. Friedlander and D. Serre. In: Vol. 4 of Handbook of Mathematical Fluid Dynamics. Elsevier (2006). [Google Scholar]
  28. J. Maxwell, On the dynamical theory of gases. Philos. Trans. R. Soc. A 157 (1866) 26–78. [Google Scholar]
  29. G. Minty, On a “monotonicity’’ method for the solution of nonlinear equations in Banach spaces. Proc. Natl. Acad. Sci. U.S.A. 50 (1963) 1038–1041. [CrossRef] [Google Scholar]
  30. J. Oldroyd, On the formulation of rheological equations of state. Proc. R. Soc. London Ser. A 200 (1950) 523–591. [CrossRef] [MathSciNet] [Google Scholar]
  31. D. Peaceman and H. Rachford, The numerical solution of parabolic elliptic differential equations. J. Soc. Indust. Appl. Math. 3 (1955) 28–41. [CrossRef] [MathSciNet] [Google Scholar]
  32. T. Perlácová and V. Prǔša, Tensorial implicit constitutive relations in mechanics of incompressible non-Newtonian fluids. J. Non-Newtonian Fluid Mech. 216 (2015) 13–21. [CrossRef] [Google Scholar]
  33. V. Prǔša, J. Málek and K. Rajagopal, Generalizations of the Navier-Stokes fluid from a new perspective. Int. J. Eng. Sci. 48 (2010) 1907–1924. [CrossRef] [Google Scholar]
  34. K. Rajagopal, On implicit constitutive theories. Appl. Math. 48 (2003) 279–319. [CrossRef] [Google Scholar]
  35. K. Rajagopal, On implicit constitutive theories for fluids. J. Fluid Mech. 550 (2006) 243–249. [CrossRef] [MathSciNet] [Google Scholar]
  36. K. Rajagopal, A new development and interpretation of the Navier-Stokes fluid which reveals why the Stokes Assumption is inapt. Int. J. Non-Linear Mech. 50 (2013) 141–151. [CrossRef] [Google Scholar]
  37. K. Rajagopal, Remarks on the notion of “pressure’’. Int. J. Non-Linear Mech. 71 (2015) 165–172. [Google Scholar]
  38. B. Rivière, Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations: Theory and Implementation. SIAM (2008). [Google Scholar]
  39. C. Le Roux and K. Rajagopal, Shear flows of a new class of power law fluids. Appl. Math. 58 (2013) 153–177. [CrossRef] [Google Scholar]
  40. G. Stokes, On the theories of internal friction of fluids in motion, and of the equilibrium and motion of elastic solids. Trans. Cambridge Phil. Soc. 8 (1845) 287–341. [Google Scholar]
  41. R. Temam, Theory and Numerical Analysis of the Navier-Stokes Equations. North-Holland, Amsterdam, The Netherlands (1977). [Google Scholar]
  42. S. Zhang, Subtetrahedral test for the positive Jacobian of hexahedral elements. Preprint (2005). [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