We propose a new full discretization of the Biot's equations in poroelasticity. The construction is driven by the inf-sup theory, which we recently developed. It builds upon the four-field formulation of the equations obtained by introducing the total pressure and the total fluid content. We discretize in space with Lagrange finite elements and in time with backward Euler. We establish inf-sup stability and quasi-optimality of the proposed discretization, with robust constants with respect to all material parameters. We further construct an interpolant showing how the error decays for smooth solutions.

翻译：我们提出了一种新的多孔弹性Biot方程全离散格式。该构造基于我们近期发展的inf-sup理论驱动，建立在通过引入总压力和总流体含量获得的四场方程表述基础上。空间离散采用Lagrange有限元，时间离散采用向后Euler法。我们证明了所提离散格式的inf-sup稳定性和拟最优性，其常数对所有材料参数具有鲁棒性。进一步构造的插值算子揭示了光滑解情况下的误差衰减规律。