We consider the numerical behavior of the fixed-stress splitting method for coupled poromechanics as undrained regimes are approached. We explain that pressure stability is related to the splitting error of the scheme, not the fact that the discrete saddle point matrix never appears in the fixed-stress approach. This observation reconciles previous results regarding the pressure stability of the splitting method. Using examples of compositional poromechanics with application to geological CO$_2$ sequestration, we see that solutions obtained using the fixed-stress scheme with a low order finite element-finite volume discretization which is not inherently inf-sup stable can exhibit the same pressure oscillations obtained with the corresponding fully implicit scheme. Moreover, pressure jump stabilization can effectively remove these spurious oscillations in the fixed-stress setting, while also improving the efficiency of the scheme in terms of the number of iterations required at every time step to reach convergence.

翻译：我们研究了在接近不排水状态时耦合孔隙力学中固定应力分裂方法的数值行为。我们解释了压力稳定性与方案的分裂误差相关，而非固定应力方法中从未出现离散鞍点矩阵这一事实。这一观察结果调和了先前关于分裂方法压力稳定性的结论。通过应用于地质CO₂封存的组分孔隙力学实例，我们发现，采用非固有inf-sup稳定的低阶有限元-有限体积离散格式的固定应力方案，其解可能表现出与相应全隐式方案相同的压力振荡现象。此外，压力跳跃稳定化能够有效消除固定应力框架中的这些伪振荡，同时通过减少每个时间步达到收敛所需的迭代次数来提高方案效率。