Locking analysis and high-order locking-free finite element method for three-dimensional electroporoelasticity equations

Electroporoelasticity equations couple Maxwell's equations with Biot's poroelasticity model and admit severe Poisson locking in standard conforming finite element discretizations when the Lam\'e constant is large. In this paper, we provide a rigorous analysis of the Poisson locking phenomenon for three-dimensional quasi-static electroporoelasticity equations and show that the spatial convergence order of conforming finite element approximations is reduced in the nearly incompressible regime. To eliminate this locking effect, we introduce a five-field formulation and a fully discrete high-order finite element method. We prove uniform stability with respect to large Lam\'e constant. Based on this, we establish a locking-free scheme by deriving its uniform error estimates with respect to the Lam\'e coefficient. The analysis covers the fully coupled electromagnetic-poroelastic system and applies to high-order elements in three dimensions. Extensive numerical experiments are presented to verify the theoretical convergence rates and demonstrate robustness with respect to the Lam\'e parameter.