Virtual element methods for the three-field formulation of time-dependent linear poroelasticity

Virtual Element Approximations for Two Species Model of the Advection-Diffusion-Reaction in Poroelastic Media

“…By extending the analysis of [12,29], we have discussed and analyzed the lowest order conforming virtual element methods for the approximation of coupled poroelasticity and ADR equations. The major contributions of this article are: showing the well-posedness of fully discrete schemes and establishing the optimal a-priori error estimates for all the variables that naturally appeared in the weak formulation.…”

“…Proof. The linear problem (3.2) in the form of an uncoupled fully discrete scheme for poroelasticity problem is well-posed for a given data and can be referred from [12]. We will ensure the existence of a unique solution of linear uncoupled ADR equation (3.3) by virtue of the Lax-Milgram lemma.…”

“…We remark that convergence analysis of VEM can be carried out similarly as in finite element methods by introducing certain projection operators onto the space of polynomial functions. Recently, VEM has been also developed in [12,14,24,25,26] for the Biot's equation and for nonlinear problems in [9,13].…”

“…0,K , where c * , c * , c * , c * , ĉ * , ĉ * are constants independent of h K . With the help of the stability of the projection operators and stabilizations terms, we obtain the following lemma ( for details, see[6,9,12] and references within).Lemma 1. For all u s h , v s h ∈ V h , p f h , q f h ∈ Q h , w h , s h ∈ W h , we have, for i = 1, 2, a h 1 (u s h , v s h ) ≤ 2µα * ∥ε(u s h )∥ 0,Ω ∥ε(v s h )∥ 0,Ω , ãh 2 (p f h , q f h ) ≤ α * c 0 + α 2 λ p f h 0,Ω q f h 0,Ω , a h 2 (p f h , q f h ) ≤ α * κ 2 η −1 ∇p f h 0,Ω ∇q f h 0,Ω , a h 3+i (w h , s h ) ≤ α * D b i ∥∇w h ∥ 0,Ω ∥∇s h ∥ 0,Ω , F h b,r (w 1,h , w 2,h ; v s h ) ≲ (ρ ∥b∥ 0,Ω ∥v s h ∥ 0,Ω + τ C k,2 ∥r h ∥ 0,Ω ∥ε(v s h )∥ 0,Ω ), m h (w h , s h ) ≤ ∥w h ∥ 0,Ω ∥s h ∥ 0,Ω , G h ℓ (q f h ) ≤ ∥ℓ∥ 0,Ω q f h 0,Ω , c h (v s h ; w h , s h ) ≲ |v s h | 1,Ω ∥w h ∥ 1,Ω ∥s h ∥ 1,Ω , J h z (w 1,h , w 2,h , u s h ; s h ) ≤ ∥z h ∥ 0,Ω ∥s h ∥ 0,Ω .…”

Parameter-robust methods for the Biot-Stokes interfacial coupling without Lagrange multipliers

“…In [12,16] (and starting from the Biot-Stokes equations advanced in [5,17]) the authors rewrite the poroelasticity equations using displacement, fluid pressure and total pressure (also as in the poromechanics formulations from [18,19,20]). Since fluid pressure in the poroelastic domain has sufficient regularity, no Lagrange multipliers are needed to enforce the coupling conditions, which resembles the different formulations for Stokes-Darcy advanced in [21,22,23,24].…”