Virtual element methods for Biot-Kirchhoff poroelasticity

TL;DR

This work develops conforming and nonconforming virtual element discretizations for the plate Biot poroelastic system, coupling transverse deflection and fluid pressure through a fourth-order and a second-order PDE. It introduces enrichment and companion operators to connect nonconforming VE spaces to conforming Sobolev spaces, and establishes a priori best-approximation error bounds alongside residual-based a posteriori estimators that are robust to the parameters $\alpha$, $\beta$, $\gamma$. Theoretical results are complemented by numerical experiments on general polygonal meshes, confirming optimal convergence and effective adaptive refinement for both smooth and non-smooth solutions. The methods offer a flexible and efficient framework for polygonal discretizations of coupled poroelastic plate problems and provide building blocks for more complex multi-layer and multiphysics models.

Abstract

This paper analyses conforming and nonconforming virtual element formulations of arbitrary polynomial degrees on general polygonal meshes for the coupling of solid and fluid phases in deformable porous plates. The governing equations consist of one fourth-order equation for the transverse displacement of the middle surface coupled with a second-order equation for the pressure head relative to the solid with mixed boundary conditions. We propose novel enrichment operators that connect nonconforming virtual element spaces of general degree to continuous Sobolev spaces. These operators satisfy additional orthogonal and best-approximation properties (referred to as a conforming companion operator in the context of finite element methods), which play an important role in the nonconforming methods. This paper proves a priori error estimates in the best-approximation form, and derives residual--based reliable and efficient a posteriori error estimates in appropriate norms, and shows that these error bounds are robust with respect to the main model parameters. The computational examples illustrate the numerical behaviour of the suggested virtual element discretisations and confirm the theoretical findings on different polygonal meshes with mixed boundary conditions.

Figures (10)

• Figure 1: Sample of pentagonal element with vertices $V_1,\dots,V_5$, edges $e_1,\dots,e_5$, corners $z_1,\dots,z_4$, and sides $s_1,\dots,s_4$.
• Figure 2: Sketch of a polygonal domain $K$ and three consecutive vertices $A,B,C$. The unit vectors $\boldsymbol{t}_1,\boldsymbol{t}_2$ form an angle $\theta$ on $A$.
• Figure 3: Approximation $u_h$ of displacement $u$ for $k=2$ (left) and $p_h$ of pressure $p$ for $\ell=1$ (right) on a smooth Voronoi mesh of 400 elements.
• Figure 4: Left (resp. right) panel displays NDof vs error in energy norm (resp. error estimator) in both uniform and adaptive refinements for conforming VEM.
• Figure 5: Left (resp. right) panel displays NDof vs error in energy norm (resp. error estimator) in both uniform and adaptive refinements for nonconforming VEM.

Theorems & Definitions (49)

• Theorem 2.1
• proof
• Remark 2.2: Simply supported boundary condition
• Proposition 3.1: Polynomial approximation brenner2008
• Theorem 3.1
• proof
• Proposition 4.1: Conforming interpolation cangiani17chen2022
• Theorem 4.1
• proof
• Theorem 5.1