A locking-free virtual element method for the three-dimensional quasi-static electroporoelasticity equations

We develop and analyze a locking-free virtual element method for the three-dimensional quasi-static electroporoelasticity equations. The pressure, electric, magnetic, and displacement fields are discretized by the lowest-order nodal, edge, face, and Stokes-like virtual element spaces, respectively, so that the scheme works on general polyhedral meshes, while the temporal discretization is carried out by a backward Euler scheme. The locking phenomenon is overcome by defining the Stokes-like space on a refined mesh ${\tilde{\mathcal{T}}}_h$, on which we can construct an interpolation operator ${\mathrm{I}}_s$ satisfying $÷({\mathrm{I}}_s\mathit{v}){\mathrm{\mid }}_K={\mathrm{\Pi }}_0^K÷\mathit{v}$, which renders the error estimates independent of the Lamé constant $\lambda$. Moreover, since the exact sequence $\text{curl}{\mathit{V}}_h^e={\mathit{V}}_{h,0}^f$ holds, the magnetic field is updated explicitly, reducing the four-field system to a three-field one and lowering the computational cost. Numerical experiments are provided to demonstrate the performance of the proposed method.