Fully Discrete Numerical Methods for Flow in Deformable Porous Media
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We consider a linearised model, in the context of mixture theory, for the flow of a fluid through a deformable porous elastic solid, where, with respect to the fluid, the solid is undergoing small but not negligible velocity. Albeit linear, the corresponding system of equations is fairly complex because it is coupled to the solid and fluid velocities. In [1], we developed the theoretical analysis of the system and proved the uniform stability of the discrete solution, considering the backward Euler method for the time discretisation, Taylor-Hood elements for the flow and piecewise quadratic elements for the solid. In this presentation, we focus on the error analysis and implementation of two fully discrete schemes: one using the first-order backward Euler method for time discretisation and the other employing the second-order BDF2 scheme. We derive error estimates for both procedures and provide numerical experiments on the corresponding non-dimensional system. Numerical experiments confirm the scheme’s performance and the model’s validity.
