Free Access
Issue
ESAIM: M2AN
Volume 55, 2021
Regular articles published in advance of the transition of the journal to Subscribe to Open (S2O). Free supplement sponsored by the Fonds National pour la Science Ouverte
Page(s) S475 - S506
DOI https://doi.org/10.1051/m2an/2020045
Published online 26 February 2021
  1. 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]
  2. E. Ahmed, J.M. Nordbotten and F.A. Radu, Adaptive asynchronous time-stepping, stopping criteria, and a posteriori error estimates for fixed-stress iterative schemes for coupled poromechanics problems. J. Comput. Appl. Math. 364 (2020) 112312. [Google Scholar]
  3. M.S. Alnæs, J. Blechta, J. Hake, A. Johansson, B. Kehlet, A. Logg, C. Richardson, J. Ring, M.E. Rognes and G.N. Wells, The fenics project version 1.5. Arch. Numer. Softw. 3 (2015) 9–23. [Google Scholar]
  4. M. Alvarez, G.N. Gatica and R. Ruiz-Baier, A posteriori error analysis for a viscous flow-transport problem. ESAIM: M2AN 50 (2016) 1789–1816. [EDP Sciences] [Google Scholar]
  5. P.R. Amestoy, I.S. Duff and J.-Y. L’Excellent, Multifrontal parallel distributed symmetric and unsymmetric solvers. Comput. Methods Appl. Mech. Eng. 184 (2000) 501–520. [Google Scholar]
  6. 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]
  7. D.N. Arnold, An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal. 19 (1982) 742–760. [Google Scholar]
  8. D.N. Arnold, F. Brezzi and M. Fortin, A stable finite element for the Stokes equations. Calcolo 21 (1984) 337–344. [Google Scholar]
  9. I. Babuška and M. Suri, Locking effects in the finite element approximation of elasticity problems. Numer. Math. 62 (1992) 439–463. [Google Scholar]
  10. M. Bause, F.A. Radu and U. Köcher, Space-time finite element approximation of the Biot poroelasticity system with iterative coupling. Comput. Methods Appl. Mech. Eng. 320 (2017) 745–768. [Google Scholar]
  11. P.J. Basser, Interstitial pressure, volume, and flow during infusion into brain tissue. Microvasc. Res. 44 (1992) 143–165. [Google Scholar]
  12. 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]
  13. M.A. Biot, General theory of three-dimensional consolidation. J. Appl. Phys. 12 (1941) 155–164. [Google Scholar]
  14. M.A. Biot, Theory of elasticity and consolidation for a porous anisotropic solid. J. Appl. Phys. 26 (1955) 182–185. [Google Scholar]
  15. D. Boffi, Three-dimensional finite element methods for the Stokes problem. SIAM J. Numer. Anal. 34 (1997) 664–670. [Google Scholar]
  16. D. Boffi, F. Brezzi, L.F. Demkowicz, R.G. Durán, R.S. Falk and M. Fortin, Mixed Finite Elements, Compatibility Conditions, and Applications, Lectures given at the C.I.M.E. Summer School held in Cetraro, June 26–July 1. Edited by Boffi and Lucia Gastaldi. Lecture Notes in Mathematics, 1939. Springer-Verlag, Berlin; Fondazione C.I.M.E., Florence (2008). [Google Scholar]
  17. D. Boffi, F. Brezzi and M. Fortin, Mixed Finite Element Methods and Applications. In: Vol. 44 of Springer Series in Computational Mathematics. Springer, Heidelberg (2013). [Google Scholar]
  18. H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations. Universitext. Springer, New York (2011). [Google Scholar]
  19. F. Brezzi and R.S. Falk, Stability of higher-order Hood-Taylor methods. SIAM J. Numer. Anal. 28 (1991) 581–590. [Google Scholar]
  20. F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. In: Vol. 15 of Springer Series in Computational Mathematics. Springer-Verlag, New York (1991). [Google Scholar]
  21. C. Carstensen, A posteriori error estimate for the mixed finite element method. Math. Comput. 66 (1997) 465–476. [Google Scholar]
  22. C. Carstensen and G. Dolzmann, A posteriori error estimates for mixed FEM in elasticity. Numer. Math. 81 (1998) 187–209. [Google Scholar]
  23. S. Caucao, D. Mora and R. Oyarzúa, A priori and a posteriori error analysis of a pseudostress-based mixed formulation of the Stokes problem with varying density. IMA J. Numer. Anal. 36 (2016) 947–983. [Google Scholar]
  24. 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]
  25. Z. Chen, Y. Xu and J. Zhang, A second-order hybrid finite volume method for solving the Stokes equation. Appl. Numer. Math. 119 (2017) 213–224. [Google Scholar]
  26. P.G. Ciarlet, The Finite Element Method for Elliptic Problems. In: Vol. 4 of Studies in Mathematics and its Applications. North-Holland Publishing Co., Amsterdam-New York-Oxford (1978). [Google Scholar]
  27. P. Clément, Approximation by finite element functions using local regularization. Rev. Française Automat. Informat. Recherche Opérationnelle Sér. 9 (1975) 77–84. [Google Scholar]
  28. C. Domnguez, G.N. Gatica and S. Meddahi, A posteriori error analysis of a fully-mixed finite element method for a two-dimensional fluid-solid interaction problem. J. Comput. Math. 33 (2015) 606–641. [Google Scholar]
  29. A. Ern and S. Meunier, A posteriori error analysis of Euler-Galerkin approximations to coupled elliptic-parabolic problems. ESAIM: M2AN 43 (2009) 353–375. [CrossRef] [EDP Sciences] [Google Scholar]
  30. Q. Fang, Mesh-based monte carlo method using fast ray-tracing in plücker coordinates. Biomed. Opt. Express 1 (2010) 165–175. [PubMed] [Google Scholar]
  31. Z.E.A. Fellah, N. Sebaa, M. Fellah, E. Ogam, F.G. Mitri, C. Depollier and W. Lauriks, Application of the Biot model to ultrasound in bone: inverse problem. IEEE Trans. Ultrason. Ferroelectr. Freq. Control 55 (2008) 1516–1523. [PubMed] [Google Scholar]
  32. F.J. Gaspar, F.J. Lisbona and C.W. Oosterlee, A stabilized difference scheme for deformable porous media and its numerical resolution by multigrid methods. Comput. Visual. Sci. 11 (2008) 67–76. [Google Scholar]
  33. G.N. Gatica, A Simple Introduction to the Mixed Finite Element Method. Theory and Applications. Springer Briefs in Mathematics. Springer, Cham (2014). [Google Scholar]
  34. G.N. Gatica, A note on stable helmholtz decompositions in 3D. Appl. Anal. 99 (2018) 1110–1121. [Google Scholar]
  35. G.N. Gatica, A. Márquez and M.A. Sánchez, Analysis of a velocity-pressure-pseudostress formulation for the stationary Stokes equations. Comput. Methods Appl. Mech. Eng. 199 (2010) 1064–1079. [Google Scholar]
  36. G.N. Gatica, R. Oyarzúa and F.-J. Sayas, A residual-based a posteriori error estimator for a fully-mixed formulation of the Stokes-Darcy coupled problem. Comput. Methods Appl. Mech. Eng. 200 (2011) 1877–1891. [Google Scholar]
  37. 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]
  38. L. Guo, J.C. Vardakis, T. Lassila, M. Mitolo, N. Ravikumar, D. Chou, M. Lange, A. Sarrami-Foroushani, B.J. Tully, Z.A. Taylor and S. Varma, Subject-specific multi-poroelastic model for exploring the risk factors associated with the early stages of Alzheimer’s disease. Interface Focus 8 (2017) 20170019. [PubMed] [Google Scholar]
  39. G. Jayaraman, Water transport in the arterial wall – a theoretical study. J. Biomech. 16 (1983) 833–840. [PubMed] [Google Scholar]
  40. J.-M. Kim and R.R. Parizek, Three-dimensional finite element modelling for consolidation due to groundwater withdrawal in a desaturating anisotropic aquifer system. Int. J. Numer. Anal. Methods Geomech. 23 (1999) 549–571. [Google Scholar]
  41. S. Kumar, R. Oyarzúa, R. Ruiz-Baier and R. Sandilya, Conservative discontinuous finite volume and mixed schemes for a new four-field formulation in poroelasticity. ESAIM: M2AN 54 (2020) 273–299. [EDP Sciences] [Google Scholar]
  42. 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]
  43. J.J. Lee, E. Piersanti, 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]
  44. X. Li, H. von Holst and S. Kleiven, Influences of brain tissue poroelastic constants on intracranial pressure (ICP) during constant-rate infusion. Comput. Methods Biomech. Biomed. Eng. 16 (2013) 1330–1343. [Google Scholar]
  45. M.A. Murad and A.F.D. Loula, On stability and convergence of finite element approximations of Biot’s consolidation problem. Int. J. Numer. Methods Eng. 37 (1994) 645–667. [Google Scholar]
  46. R. Oyarzúa and R. Ruiz-Baier, Locking-free finite element methods for poroelasticity. SIAM J. Numer. Anal. 54 (2016) 2951–2973. [Google Scholar]
  47. 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]
  48. P.J. Phillips and M.F. Wheeler, Overcoming the problem of locking in linear elasticity and poroelasticity: an heuristic approach. Comput. Geosci. 13 (2009) 5–12. [Google Scholar]
  49. A. Plaza and G.F. Carey, Local refinement of simplicial grids based on the skeleton. Appl. Numer. Math. 32 (2000) 195–218. [Google Scholar]
  50. 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]
  51. 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]
  52. R.E. Showalter, Diffusion in poro-elastic media. J. Math. Anal. Appl. 251 (2000) 310–340. [Google Scholar]
  53. R.E. Showalter, Diffusion in deformable media. In: Vol. 131 of Resource Recovery, Confinement, and Remediation of Environmental Hazards (Minneapolis, MN, 2000). The IMA Volumes in Mathematics and its Applications. Springer, New York (2002) 115–129. [Google Scholar]
  54. K. Terzaghi, Principle of soil mechanics. Engineering News Record, A Series of Articles (1925). [Google Scholar]
  55. B. Tully and Y. Ventikos, Cerebral water transport using multiple-network poroelastic theory: application to normal pressure hydrocephalus. J. Fluid Mech. 667 (2011) 188–215. [Google Scholar]
  56. 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. Micro and Nano Flows 2014 (MNF2014) – Biomedical Stream. Med. Eng. Phys. 38 (2016) 48–57. [CrossRef] [PubMed] [Google Scholar]
  57. R. Verfürth, A posteriori error estimation and adaptive mesh-refinement techniques. J. Comput. Appl. Math. 50 (1994) 67–83. [Google Scholar]
  58. R. Verfürth, A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Wiley Teubner, Chichester (1996). [Google Scholar]
  59. R. Verfürth, A review of a posteriori error estimation techniques for elasticity problems. Comput. Methods Appl. Mech. Eng. 176 (1999) 419–440. [Google Scholar]
  60. M. Wangen, S. Gasda and T. Bjørnarå, Geomechanical consequences of large-scale fluid storage in the Utsira Formation in the North Sea. Energy Proc. 97 (2016) 486–493. [Google Scholar]
  61. M. 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]
  62. 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]
  63. S.-Y. Yi, A coupling of nonconforming and mixed finite element methods for Biot’s consolidation model. Numer. Methods Part. Differ. Equ. 29 (2013) 1749–1777. [Google Scholar]
  64. S.-Y. Yi, Convergence analysis of a new mixed finite element method for Biot’s consolidation model. Numer. Methods Part. Differ. Equ. 30 (2014) 1189–1210. [Google Scholar]
  65. S.-Y. Yi, A study of two modes of locking in poroelasticity. SIAM J. Numer. Anal. 55 (2017) 1915–1936. [Google Scholar]
  66. J. Young, B. Rivière, C.S. Cox, Jr. and K. Uray, A mathematical model of intestinal oedema formation. Math. Med. Biol. 31 (2014) 1–15. [Google Scholar]

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