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We present a finite element discretisation to model the interaction between aporoelastic structure and an elastic medium. The consolidation problemconsiders fully coupled deformations across an interface, ensuring continuityof displacement and total traction, as well as no-flux for the fluid phase. Ourformulation of the poroelasticity equations incorporates displacement, fluidpressure, and total pressure, while the elasticity equations adopt adisplacement-pressure formulation. Notably, the transmission conditions at theinterface are enforced without the need for Lagrange multipliers. Wedemonstrate the stability and convergence of the divergence-conforming finiteelement method across various polynomial degrees. The a priori error boundsremain robust, even when considering large variations in intricate modelparameters such as Lam\'e constants, permeability, and storativity coefficient.To enhance computational efficiency and reliability, we develop residual-baseda posteriori error estimators that are independent of the aforementionedcoefficients. Additionally, we devise parameter-robust and optimal blockdiagonal preconditioners. Through numerical examples, including adaptivescenarios, we illustrate the scheme's properties such as convergence andparameter robustness.

Efficient and reliable divergence-conforming methods for an elasticity-poroelasticity interface problem