We present and analyze a hybridizable discontinuous Galerkin (HDG) finite element method for the coupled Stokes--Biot problem. Of particular interest is that the discrete velocities and displacement are $H(\text{div})$-conforming and satisfy the compressibility equations pointwise on the elements. Furthermore, in the incompressible limit, the discretization is strongly conservative. We prove well-posedness of the discretization and, after combining the HDG method with backward Euler time stepping, present a priori error estimates that demonstrate that the method is free of volumetric locking. Numerical examples further demonstrate optimal rates of convergence in the $L^2$-norm for all unknowns and that the discretization is locking-free.

翻译：本文提出并分析了一种用于耦合Stokes–Biot问题的混合化间断伽辽金(HDG)有限元方法。特别值得关注的是，离散速度和位移满足$H(\text{div})$协调性，并且在单元上逐点满足可压缩性方程。此外，在不可压缩极限下，该离散格式具有强守恒性。我们证明了离散格式的适定性，并在将HDG方法与向后欧拉时间推进相结合后，给出了先验误差估计，表明该方法不受体积锁定的影响。数值算例进一步展示了所有未知量在$L^2$范数下的最优收敛速率，并验证了该离散格式无锁定现象。