Virtual element approximations for two species model of the Advection-Diffusion-Reaction in poroelastic media
Abstract
This paper proposes virtual element methods for approximating the mathematical model consisting of coupled poroelastic and Advection-Diffusion-Reaction (ADR) equations. The space discretization relies on virtual element spaces containing piecewise linear polynomials as well as non-polynomials for displacement, pressure and concentrations, and piecewise constant for total pressure; a backwardEuler scheme is employed for the approximation of time derivative. Using standard techniques of explicit schemes, the well-posedness of the resultant fully discrete scheme is derived. Moreover, under certain regularity assumptions on the mesh, optimal apriori error estimates are established by introducing suitable projection operators. Several numerical experiments are presented to validate the theoretical convergence rate and exhibit the proposed scheme’s performance.
Keyword : poroelastic equation, advection-diffusion-reaction equation, virtual element method, lowest order, inf-sup condition, numerical experiments, convergence analysis
How to Cite
Verma, N., & Kumar, S. (2022). Virtual element approximations for two species model of the Advection-Diffusion-Reaction in poroelastic media. Mathematical Modelling and Analysis, 27(4), 668–690. https://doi.org/10.3846/mma.2022.15542
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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J. Lee, E. Piersanti, K.-A. Mardal and M.E. Rognes. A mixed finite element method for nearly incompressible multiple-network poroelasticity. SIAM Journal on Scientific Computing, 41(2):A722–A747, 2019. https://doi.org/10.1137/18M1182395
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A. Sreekumar, S. P. Triantafyllou, F.-X. B´ecot and F. Chevillotte. Multiscale VEM for the Biot consolidation analysis of complex and highly heterogeneous domains. Computer Methods in Applied Mechanics and Engineering, 375:113543, 2021. https://doi.org/10.1016/j.cma.2020.113543
A. Sreekumar, S. P. Triantafyllou and F. Chevillotte. Virtual elements for sound propagation in complex poroelastic media. Computational Mechanics, 144:347– 382, 2021. https://doi.org/10.1007/s00466-021-02078-2
X. Tang, Z. Liu, B. Zhang and M. Feng. On the locking-free threefield virtual element methods for Biot’s consolidation model in poroelasticity. ESAIM. Mathematical Modelling and Numerical Analysis, 55:S909–S939, 2021. https://doi.org/10.1051/m2an/2020064
G. Vacca and L.B. da Veiga. Virtual element methods for parabolic problems on polygonal meshes. Numerical Methods for Partial Differential Equations. An International Journal, 31(6):2110–2134, 2015. https://doi.org/10.1002/num.21982
N. Verma, B. Gómez-Vargas, L. Miguel De Oliveira Vilaca, S. Kumar and R. Ruiz-Baier. Well-posedness and discrete analysis for advection-diffusionreaction in poroelastic media. Applicable Analysis, 101(14):4914–4941, 2022. https://doi.org/10.1080/00036811.2021.1877677
N. Verma and S. Kumar. Lowest order virtual element approximations for transient Stokes problem on polygonal meshes. Calcolo, 58(48), 2021. https://doi.org/10.1007/s10092-021-00440-7
S.-Y. Yi. A study of two modes of locking in poroelasticity. SIAM Journal on Numerical Analysis, 55(4):1915–1936, 2017. https://doi.org/10.1137/16M1056109
V. Anaya, M. Bendahmane, D. Mora and R. Ruiz Baier. On a vorticity-based formulation for reaction-diffusion-Brinkman systems. Networks and Heterogeneous Media, 13(1):69–94, 2018. https://doi.org/10.3934/nhm.2018004
V. Anaya, Z. de Wijn, B. Gómez-Vargas, D. Mora and R. Ruiz-Baier. Rotation-based mixed formulations for an elasticity-poroelasticity interface problem. SIAM Journal on Scientific Computing, 42(1):B225–B249, 2020. https://doi.org/10.1137/19M1268343
P.F. Antonietti, L. Beirão da Veiga, D. Mora and M. Verani. A stream virtual element formulation of the Stokes problem on polygonal meshes. SIAM Journal on Numerical Analysis, 52(1):386–404, 2014. https://doi.org/10.1137/13091141X
Beirão da Veiga L., Brezzi F., A. Cangiani, G. Manzini, L.D. Marini and A. Russo. Basic principles of virtual element methods. Mathematical Models and Methods in Applied Sciences, 23(1):199–214, 2013. https://doi.org/10.1142/S0218202512500492
L. Beirão da Veiga, F. Brezzi, L.D. Marini and A. Russo. Virtual element implementation for general elliptic equations. In Building bridges: connections and challenges in modern approaches to numerical partial differential equations, volume 114 of Lecture Notes in Computational Science and Engineering, pp. 39–71. Springer, 2016. https://doi.org/10.1007/978-3-319-41640-3_2
L. Beirão da Veiga, C. Lovadina and G. Vacca. Divergence free virtual elements for the Stokes problem on polygonal meshes. ESAIM. Mathematical Modelling and Numerical Analysis, 51(2):509–535, 2017. https://doi.org/10.1051/m2an/2016032
L. Beirão da Veiga, C. Lovadina and G. Vacca. Virtual elements for the Navier– Stokes problem on polygonal meshes. SIAM Journal on Numerical Analysis, 56(3):1210–1242, 2018. https://doi.org/10.1137/17M1132811
L. Beirão da Veiga, A. Pichler and G. Vacca. A virtual element method for the miscible displacement of incompressible fluids in porous media. Computer Methods in Applied Mechanics and Engineering, 375:113649, 2021. https://doi.org/10.1016/j.cma.2020.113649
D. Boffi, M. Botti and D. Di Pietro. A nonconforming high-order method for the Biot problem on general meshes. SIAM Journal on Scientific Computing, 38(3):A1508–A1537, 2016.
M. Botti, D. Di Pietro and P. Sochala. A hybrid high-order discretization method for nonlinear poroelasticity. Computational Methods in Applied Mathematics, 20(2):227–249, 2020. https://doi.org/10.1515/cmam-2018-0142
R. Bürger, S. Kumar, D. Mora, R. Ruiz-Baier and N. Verma. Virtual element methods for the three-field formulation of time-dependent linear poroelasticity. Advances in Computational Mathematics volume, 47(2), 2021. https://doi.org/10.1007/s10444-020-09826-7
A. Cangiani, P. Chatzipantelidis, G. Diwan and E.H. Georgoulis. Virtual element method for quasilinear elliptic problems. IMA Journal of Numerical Analysis, 40(4):2450–2472, 2020. https://doi.org/10.1093/imanum/drz035
J. Coulet, I. Faille, V. Girault, N. Guy and F. Nataf. A fully coupled scheme using virtual element method and finite volume for poroelasticity. Computational Geosciences,24(2):381–403,2020. https://doi.org/10.1007/s10596-019-09831-w
L.M. De Oliveira Vilaca, B. Gómez-Vargas, S. Kumar, R. Ruiz-Baier and N. Verma. Stability analysis for a new model of multi-species convectiondiffusion-reaction in poroelastic tissue. Applied Mathematical Modelling. Simulation and Computation for Engineering and Environmental Systems, 84:425–446, 2020. https://doi.org/10.1016/j.apm.2020.04.014
V. Girault, G. Pencheva, M.F. Wheeler and T. Wildey. Domain decomposition for poroelasticity and elasticity with DG jumps and mortars. Mathematical Models and Methods in Applied Sciences, 21(1):169–213, 2011. https://doi.org/10.1142/S0218202511005039
V. Girault and P.-A. Raviart. Finite element approximation of the NavierStokes equations. Lecture Notes in Mathematics. Springer-Verlag, 1979. https://doi.org/10.1007/BFb0063447
J. Heywood and R. Rannacher. Finite-element approximation of the nonstationary Navier-Stokes problem. IV. Error analysis for second-order time discretization. SIAM Journal on Numerical Analysis, 27(2):353–384, 1990. https://doi.org/10.1137/0727022
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. Mathematical Modelling and Numerical Analysis, 54(1):273–299, 2020. https://doi.org/10.1051/m2an/2019063
J. Lee, K. Mardal and R. Winther. Parameter-robust discretization and preconditioning of Biot’s consolidation model. SIAM Journal on Scientific Computing, 39(1):A1–A24, 2017. https://doi.org/10.1137/15M1029473
J. Lee, E. Piersanti, K.-A. Mardal and M.E. Rognes. A mixed finite element method for nearly incompressible multiple-network poroelasticity. SIAM Journal on Scientific Computing, 41(2):A722–A747, 2019. https://doi.org/10.1137/18M1182395
R. Oyarzúa and R. Ruiz-Baier. Locking-free finite element methods for poroelasticity. SIAM Journal on Numerical Analysis, 54(5):2951–2973, 2016. https://doi.org/10.1137/15M1050082
B. Rivière, J. Tan and T. Thompson. Error analysis of primal discontinuous Galerkin methods for a mixed formulation of the Biot equations. Computers & Mathematics with Applications. An International Journal, 73(4):666–683, 2017. https://doi.org/10.1016/j.camwa.2016.12.030
A. Sreekumar, S. P. Triantafyllou, F.-X. B´ecot and F. Chevillotte. Multiscale VEM for the Biot consolidation analysis of complex and highly heterogeneous domains. Computer Methods in Applied Mechanics and Engineering, 375:113543, 2021. https://doi.org/10.1016/j.cma.2020.113543
A. Sreekumar, S. P. Triantafyllou and F. Chevillotte. Virtual elements for sound propagation in complex poroelastic media. Computational Mechanics, 144:347– 382, 2021. https://doi.org/10.1007/s00466-021-02078-2
X. Tang, Z. Liu, B. Zhang and M. Feng. On the locking-free threefield virtual element methods for Biot’s consolidation model in poroelasticity. ESAIM. Mathematical Modelling and Numerical Analysis, 55:S909–S939, 2021. https://doi.org/10.1051/m2an/2020064
G. Vacca and L.B. da Veiga. Virtual element methods for parabolic problems on polygonal meshes. Numerical Methods for Partial Differential Equations. An International Journal, 31(6):2110–2134, 2015. https://doi.org/10.1002/num.21982
N. Verma, B. Gómez-Vargas, L. Miguel De Oliveira Vilaca, S. Kumar and R. Ruiz-Baier. Well-posedness and discrete analysis for advection-diffusionreaction in poroelastic media. Applicable Analysis, 101(14):4914–4941, 2022. https://doi.org/10.1080/00036811.2021.1877677
N. Verma and S. Kumar. Lowest order virtual element approximations for transient Stokes problem on polygonal meshes. Calcolo, 58(48), 2021. https://doi.org/10.1007/s10092-021-00440-7
S.-Y. Yi. A study of two modes of locking in poroelasticity. SIAM Journal on Numerical Analysis, 55(4):1915–1936, 2017. https://doi.org/10.1137/16M1056109