Optimal <i>L</i> 2- error estimates of a locking free numerical method for a quasi-static linear poroelasticity

Abstract The optimal $L^{2}$-error estimates of a locking-free numerical method are established for a quasi-static linear poroelasticity. Existing research shows that the original model can be reformulated as a generalized Stokes equation coupled with a diffusion problem, which inherently mitigates two locking phenomena associated with the continuous Galerkin method. However, previous studies only achieved optimal error estimates in the $H^{1}$-norm, lacking results for displacement in the $L^{2}$-norm. We find that the main barrier to achieve these estimates is arising from the influence of Lamé constant $\lambda $ on numerical schemes. By taking the value of $\lambda $ into account, we design a fully discrete mixed finite element method to solve the reformulated problem. We prove that the discrete formulation satisfies the inf-sup stability condition across various finite element pairs. Our results show that when $\lambda &amp;lt;\infty,$ the generalized Stokes problem is stable with equal-order Lagrange element pairs, making the Taylor–Hood pairs unnecessary and enabling optimal $L^{2}$-error estimates for displacement. For $\lambda \rightarrow \infty,$ the problem reduces to a standard Stokes problem. Using inf-sup bounds and the Aubin–Nitsche duality method, we provide optimal $L^{2}$-error analysis. Numerical examples are included to confirm our theoretical findings.