### Abstract

In this work, we consider the popular P1–RT0–P0 discretization of the three-field formulation of Biot's consolidation problem. Since this finite-element formulation is not uniformly stable with respect to the physical parameters, several issues arise in numerical simulations. For example, when the permeability is small with respect to the mesh size, volumetric locking may occur. To alleviate such problems, we consider a well-known stabilization technique with face bubble functions. We then design a perturbation of the bilinear form, which allows for local elimination of the bubble functions. We further prove that such perturbation is consistent and the resulting scheme has optimal approximation properties for both Biot's model as well as the Stokes’ equations. For the former, the number of degrees of freedom is the same as for the classical P1–RT0–P0 discretization and for the latter (Stokes’ equations) the number of degrees of freedom is the same as for a P1–P0 discretization. We present numerical tests confirming the theoretical results for the poroelastic and the Stokes’ test problems.

Original language | English (US) |
---|---|

Pages (from-to) | 467-484 |

Number of pages | 18 |

Journal | Computer Methods in Applied Mechanics and Engineering |

Volume | 341 |

DOIs | |

State | Published - Nov 1 2018 |

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### All Science Journal Classification (ASJC) codes

- Computational Mechanics
- Mechanics of Materials
- Mechanical Engineering
- Physics and Astronomy(all)
- Computer Science Applications

### Cite this

*Computer Methods in Applied Mechanics and Engineering*,

*341*, 467-484. https://doi.org/10.1016/j.cma.2018.07.003

}

*Computer Methods in Applied Mechanics and Engineering*, vol. 341, pp. 467-484. https://doi.org/10.1016/j.cma.2018.07.003

**New stabilized discretizations for poroelasticity and the Stokes’ equations.** / Rodrigo, C.; Hu, X.; Ohm, P.; Adler, J. H.; Gaspar, F. J.; Zikatanov, L. T.

Research output: Contribution to journal › Article

TY - JOUR

T1 - New stabilized discretizations for poroelasticity and the Stokes’ equations

AU - Rodrigo, C.

AU - Hu, X.

AU - Ohm, P.

AU - Adler, J. H.

AU - Gaspar, F. J.

AU - Zikatanov, L. T.

PY - 2018/11/1

Y1 - 2018/11/1

N2 - In this work, we consider the popular P1–RT0–P0 discretization of the three-field formulation of Biot's consolidation problem. Since this finite-element formulation is not uniformly stable with respect to the physical parameters, several issues arise in numerical simulations. For example, when the permeability is small with respect to the mesh size, volumetric locking may occur. To alleviate such problems, we consider a well-known stabilization technique with face bubble functions. We then design a perturbation of the bilinear form, which allows for local elimination of the bubble functions. We further prove that such perturbation is consistent and the resulting scheme has optimal approximation properties for both Biot's model as well as the Stokes’ equations. For the former, the number of degrees of freedom is the same as for the classical P1–RT0–P0 discretization and for the latter (Stokes’ equations) the number of degrees of freedom is the same as for a P1–P0 discretization. We present numerical tests confirming the theoretical results for the poroelastic and the Stokes’ test problems.

AB - In this work, we consider the popular P1–RT0–P0 discretization of the three-field formulation of Biot's consolidation problem. Since this finite-element formulation is not uniformly stable with respect to the physical parameters, several issues arise in numerical simulations. For example, when the permeability is small with respect to the mesh size, volumetric locking may occur. To alleviate such problems, we consider a well-known stabilization technique with face bubble functions. We then design a perturbation of the bilinear form, which allows for local elimination of the bubble functions. We further prove that such perturbation is consistent and the resulting scheme has optimal approximation properties for both Biot's model as well as the Stokes’ equations. For the former, the number of degrees of freedom is the same as for the classical P1–RT0–P0 discretization and for the latter (Stokes’ equations) the number of degrees of freedom is the same as for a P1–P0 discretization. We present numerical tests confirming the theoretical results for the poroelastic and the Stokes’ test problems.

UR - http://www.scopus.com/inward/record.url?scp=85050484423&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85050484423&partnerID=8YFLogxK

U2 - 10.1016/j.cma.2018.07.003

DO - 10.1016/j.cma.2018.07.003

M3 - Article

AN - SCOPUS:85050484423

VL - 341

SP - 467

EP - 484

JO - Computer Methods in Applied Mechanics and Engineering

JF - Computer Methods in Applied Mechanics and Engineering

SN - 0374-2830

ER -