Free Access
Volume 54, Number 1, January-February 2020
Page(s) 273 - 299
Published online 31 January 2020
  1. F. Aguilar, F.L. Gaspar and C. Rodrigo, Numerical stabilization of Biot’s consolidation model by a perturbation on the flow equation. Int. J. Numer. Methods Eng. 75 (2008) 1282–1300. [Google Scholar]
  2. E. Ahmed, F.A. Radu and J.M. Nordbotten, Adaptive poromechanics computations based on a posteriori error estimates for fully mixed formulations of Biot’s consolidation model. Comput. Methods Appl. Mech. Eng. 347 (2019) 264–294. [Google Scholar]
  3. I. Ambartsumyan, E. Khattatov, I. Yotov and P. Zunino, A Lagrange multiplier method for a Stokes-Biot fluid-poroelastic structure interaction model. Numer. Math. 140 (2018) 513–553. [Google Scholar]
  4. V. Anaya, Z. De Wijn, B. Gómez-Vargas, D. Mora and R. Ruiz-Baier, Rotation-based mixed formulations for an elasticity-poroelasticity interface problem. Submitted preprint (2019). Available from [Google Scholar]
  5. V. Anaya, Z. De Wijn, D. Mora and R. Ruiz-Baier, Mixed displacement-rotation-pressure formulations for linear elasticity. Comput. Methods Appl. Mech. Eng. 344 (2019) 71–94. [Google Scholar]
  6. D.N. Arnold, An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal. 19 (1982) 742–760. [Google Scholar]
  7. D.N. Arnold, F. Brezzi, B. Cockburn and L.D. Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39 (2002) 1749–1779. [Google Scholar]
  8. R. Asadi, B. Ataie-Ashtiani and C.T. Simmons, Finite volume coupling strategies for the solution of a Biot consolidation model. Comput. Geotech. 55 (2014) 494–505. [Google Scholar]
  9. T. Bærland, J.J. Lee, K.-A. Mardal and R. Winther, Weakly imposed symmetry and robust preconditioners for Biot’s consolidation model. Comput. Methods Appl. Math. 17 (2017) 377–396. [CrossRef] [Google Scholar]
  10. L. Berger, R. Bordas, D. Kay and S. Tavener, Stabilized lowest-order finite element approximation for linear three-field poroelasticity. SIAM J. Sci. Comput. 37 (2015) A2222–A2245. [Google Scholar]
  11. L. Berger, R. Bordas, D. Kay and S. Tavener, A stabilized finite element method for finite-strain three-field poroelasticity. Comput. Mech. 60 (2017) 51–68. [PubMed] [Google Scholar]
  12. M.A. Biot, Theory of elasticity and consolidation for a porous anisotropic solid. J. Appl. Phys. 26 (1955) 182–185. [Google Scholar]
  13. D. Boffi, M. Botti and D.A. Di Pietro, A nonconforming high-order method for the Biot problem on general meshes. SIAM J. Sci. Comput. 38 (2016) A1508–A1537. [Google Scholar]
  14. D. Boffi, F. Brezzi and M. Fortin, Mixed finite element methods and applications. In: Vol. 44 of Springer Series in Computational Mathematics. Springer (2010). [Google Scholar]
  15. H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer Verlag (2010). [CrossRef] [Google Scholar]
  16. M. Bukač, I. Yotov and P. Zunino, Dimensional model reduction for flow through fractures in poroelastic media. ESAIM: M2AN 51 (2017) 1429–1471. [EDP Sciences] [Google Scholar]
  17. R. Bürger, S. Kumar and R. Ruiz-Baier, Discontinuous finite volume element discretization for coupled flow-transport problems arising in models of sedimentation. J. Comput. Phys. 299 (2015) 446–471. [Google Scholar]
  18. C. Carstensen, N. Nataraj and A.K. Pani, Comparison results and unified analysis for first-order finite volume element methods for a Poisson model problem. IMA J. Numer. Anal. 36 (2016) 1120–1142. [CrossRef] [Google Scholar]
  19. N. Castelletto, J.A. White and M. Ferronato, Scalable algorithms for three-field mixed finite element coupled poromechanics. J. Comput. Phys. 327 (2016) 894–918. [Google Scholar]
  20. D. Chapelle and P. Moireau, General coupling of porous flows and hyperelastic formulations – From thermodynamics principles to energy balance and compatible time schemes. Eur. J. Mech. B/Fluids 46 (2014) 82–96. [CrossRef] [Google Scholar]
  21. Y. Chen, Y. Luo and M. Feng, Analysis of a discontinuous Galerkin method for the Biot’s consolidation problem. Appl. Math. Comput. 219 (2013) 9043–9056. [Google Scholar]
  22. Z. Chen, Y. Xu and Y. Zhang, A second-order hybrid finite volume method for solving the Stokes equation. Appl. Numer. Math. 119 (2017) 213–224. [Google Scholar]
  23. Z. De Wijn, R. Oyarzúa and R. Ruiz-Baier, A second-order finite-volume-element scheme for linear poroelasticity. Submitted preprint (2019). Available from [Google Scholar]
  24. Q. Deng, V. Ginting, B. McCaskill and P. Torsu, A locally conservative stabilized continuous Galerkin finite element method for two-phase flow in poroelastic subsurfaces. J. Comput. Phys. 347 (2017) 78–98. [Google Scholar]
  25. D. Di Pietro and A. Ern, Mathematical aspects of discontinuous Galerkin methods. In: Vol. 69 of Mathématiques & Applications. Springer, Heidelberg (2012). [CrossRef] [Google Scholar]
  26. Q. Fang and D.A. Boas, Monte Carlo simulation of photon migration in 3D turbid media accelerated by graphics processing units. Opt. Express 17 (2009) 20178–20190. [CrossRef] [PubMed] [Google Scholar]
  27. X. Feng, Z. Ge and Y. Li, Analysis of a multiphysics finite element method for a poroelasticity model. IMA J. Numer. Anal. 38 (2018) 330–359. [CrossRef] [Google Scholar]
  28. G. Fu, A high-order HDG method for the Biot’s consolidation model. Comput. Math. Appl. 77 (2019) 237–252. [Google Scholar]
  29. G.N. Gatica, A Simple Introduction to the Mixed Finite Element Method. Theory and Applications. Springer Briefs in Mathematics. Springer, Cham (2014). [CrossRef] [Google Scholar]
  30. C. Geuzaine and J.-F. Remacle, Gmsh: a three-dimensional finite element mesh generator with built-in pre- and post-processing facilities. Int. J. Numer. Methods Eng. 79(11) (2009) 1309–1331. [Google Scholar]
  31. V. Girault and P.-A. Raviart, Finite element approximation of the Navier–Stokes equations. In: Vol. 749 of Lecture Notes in Mathematics. Springer-Verlag, Berlin-New York (1979). [CrossRef] [Google Scholar]
  32. V. Girault, M.F. Wheeler, B. Ganis and M.E. Mear, A lubrication fracture model in a poro-elastic medium. Math. Models Methods Appl. Sci. 25 (2015) 587–645. [Google Scholar]
  33. Q. Hong and J. Kraus, Parameter-robust stability of classical three-field formulation of Biot’s consolidation model. Electron. Trans. Numer. Anal. 48 (2018) 202–226. [CrossRef] [Google Scholar]
  34. Q. Hong, J. Kraus, M. Lymbery and F. Philo, Conservative discretizations and parameter-robust preconditioners for Biot and multiple-network flux-based poroelasticity models. Numer. Linear Alg. Appl. 26 (2019) e2242. [CrossRef] [Google Scholar]
  35. X. Hu, C. Rodrigo, F.J. Gaspar and L.T. Zikatanov, A non-conforming finite element method for the Biot’s consolidation model in poroelasticity. J. Comput. Appl. Math. 310 (2017) 143–154. [Google Scholar]
  36. G. Kanschat and B. Rivière, A finite element method with strong mass conservation for Biot’s linear consolidation model. J. Sci. Comput. 77 (2018) 1762–1779. [Google Scholar]
  37. S. Kumar and R. Ruiz-Baier, Equal order discontinuous finite volume element methods for the Stokes problem. J. Sci. Comput. 65 (2015) 956–978. [Google Scholar]
  38. R. Lazarov and X. Ye, Stabilized discontinuous finite element approximations for Stokes equations. J. Comput. Appl. Math. 198 (2007) 236–252. [Google Scholar]
  39. J.J. Lee, K.-A. Mardal and R. Winther, Parameter-robust discretization and preconditioning of Biot’s consolidation model. SIAM J. Sci. Comput. 39 (2017) A1–A24. [Google Scholar]
  40. J.J. Lee, E. Persanti, K.-A. Mardal and M.E. Rognes, A mixed finite element method for nearly incompressible multiple-network poroelasticity. SIAM J. Sci. Comput. 41 (2019) A722–A747. [Google Scholar]
  41. R.W. Lewis and B.A. Schrefler, The Finite Element Method in the Deformation of and Consolidation of Porous Media. Wiley & Sons, Chichester (1987). [Google Scholar]
  42. R. Liu, M.F. Wheeler, C.N. Dawson and R.H. Dean, On a coupled discontinuous/continuous Galerkin framework and an adaptive penalty scheme for poroelasticity problems. Comput. Methods Appl. Mech. Eng. 198 (2009) 3499–3510. [Google Scholar]
  43. K.-A. Mardal and R. Winther, Preconditioning discretizations of systems of partial differential equations. Numer. Linear Algebra Appl. 18 (2011) 1–40. [Google Scholar]
  44. M.A. Murad, V. Thomée and A.F.D. Loula, Asymptotic behavior of semi discrete finite-element approximations of Biot’s consolidation problem. SIAM J. Numer. Anal. 33 (1996) 1065–1083. [Google Scholar]
  45. A. Naumovich, On finite volume discretization of the three-dimensional Biot poroelasticity system in multilayer domains. Comput. Methods Appl. Math. 6 (2006) 306–325. [CrossRef] [Google Scholar]
  46. A. Naumovich and F.J. Gaspar, On a multigrid solver for the three-dimensional Biot poroelasticity system in multilayered domains. Comput. Vis. Sci. 11 (2008) 77–87. [Google Scholar]
  47. R. Oyarzúa and R. Ruiz-Baier, Locking-free finite element methods for poroelasticity. SIAM J. Numer. Anal. 54 (2016) 2951–2973. [Google Scholar]
  48. R. Penta, D. Ambrosi and R.J. Shipley, Effective governing equations for poroelastic growing media. Q. J. Mech. Appl. Math. 67 (2014) 69–91. [Google Scholar]
  49. G.P. Peters and D.W. Smith, Solute transport through a deforming porous medium. Int. J. Numer. Anal. Methods Geomech. 26 (2002) 683–717. [Google Scholar]
  50. P.J. Phillips and M.F. Wheeler, A coupling of mixed and discontinuous Galerkin finite-element methods for poroelasticity. Comput. Geosci. 12 (2008) 417–435. [Google Scholar]
  51. M. Radszuweit, H. Engel and M. Bär, An active poroelastic model for mechanochemical patterns in protoplasmic droplets of physarum polycephalum.PLoS One 9 (2014) e99220. [CrossRef] [PubMed] [Google Scholar]
  52. R. Riedlbeck, D.A. Di Pietro, A. Ern, S. Granet and K. Kazymyrenko, Stress and flux reconstruction in Biot’s poro-elasticity problem with application to a posteriori error analysis. Comput. Math. Appl. 73 (2017) 1593–1610. [Google Scholar]
  53. B. Rivière, J. Tan and T. Thompson, Error analysis of primal discontinuous Galerkin methods for a mixed formulation of the Biot equations. Comput. Math. Appl. 73 (2017) 666–683. [Google Scholar]
  54. C. Rodrigo, F.J. Gaspar, X. Hu and L.T. Zikatanov, Stability and monotonicity for some discretizations of the Biot’s consolidation model. Comput. Methods Appl. Mech. Eng. 298 (2016) 183–204. [Google Scholar]
  55. R. Ruiz-Baier and I. Lunati, Mixed finite element – discontinuous finite volume element discretization of a general class of multicontinuum models. J. Comput. Phys. 322 (2016) 666–688. [Google Scholar]
  56. R. Sacco, P. Causin, C. Lelli and M.T. Raimondi, A poroelastic mixture model of mechanobiological processes in biomass growth: theory and application to tissue engineering. Meccanica 52 (2017) 3273–3297. [PubMed] [Google Scholar]
  57. R.E. Showalter, Diffusion in poro-elastic media. J. Math. Anal. Appl. 251 (2000) 310–340. [Google Scholar]
  58. K.H. Støverud, M. Alnæs, H.P. Langtangen, V. Haughton and K.-A. Mardal, Poro-elastic modeling of Syringomyelia – a systematic study of the effects of pia mater, central canal, median fissure, white and gray matter on pressure wave propagation and fluid movement within the cervical spinal cord. Comput. Methods Biomech. Biomed. Eng. 19 (2016) 686–698. [CrossRef] [Google Scholar]
  59. M. Sun and H. Rui, A coupling of weak Galerkin and mixed finite element methods for poroelasticity. Comput. Math. Appl. 73 (2017) 804–823. [Google Scholar]
  60. J.C. Vardakis, D. Chou, B.J. Tully, C.C. Hung, T.H. Lee, P.-H. Tsui and Y. Ventikos, Investigating cerebral oedema using poroelasticity. Med. Eng. Phys. 38 (2016) 48–57. [CrossRef] [PubMed] [Google Scholar]
  61. A.-T. Vuong, L. Yoshihara and W.A. Wall, A general approach for modeling interacting flow through porous media under finite deformations. Comput. Methods Appl. Mech. Eng. 283 (2015) 1240–1259. [Google Scholar]
  62. M.F. Wheeler, G. Xue and I. Yotov, Coupling multipoint flux mixed finite element methods with continuous Galerkin methods for poroelasticity. Comput. Geosci. 18 (2014) 57–75. [Google Scholar]
  63. J.A. White and R.I. Borja, Stabilized low-order finite elements for coupled solid-deformation/fluid-diffusion and their application to fault zone transients. Comput. Methods Appl. Mech. Eng. 197 (2008) 4353–4366. [Google Scholar]
  64. X. Ye, A discontinuous finite volume method for the Stokes problem. SIAM J. Numer. Anal. 44 (2006) 183–198. [Google Scholar]
  65. S.-Y. Yi, A Coupling of nonconforming and mixed finite element methods for Biot’s consolidation model. Numer. Methods Part. Diff. Equ. 29 (2013) 1749–1777. [CrossRef] [Google Scholar]
  66. S.-Y. Yi, Convergence analysis of a new mixed finite element method for Biot’s consolidation model. Numer. Methods Part. Diff. Equ. 30 (2014) 1189–1210. [CrossRef] [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