This work focuses on the numerical solution of the dynamics of a poroelastic material in the frequency domain. We provide a detailed stability analysis based on the application of the Fredholm alternative in the continuous case, considering a total pressure formulation of the Biot’s equations. In the discrete setting, we propose a stabilized equal order finite element method complemented by an additional pressure stabilization to enhance the robustness of the numerical scheme with respect to the fluid permeability. Utilizing the Fredholm alternative, we extend the well-posedness results to the discrete setting, obtaining theoretical optimal convergence for the case of linear finite elements. We present different numerical experiments to validate the proposed method. First, we consider model problems with known analytic solutions in two and three dimensions. As next, we show that the method is robust for a wide range of permeabilities, including the case of discontinuous coefficients. Lastly, we show the application for the simulation of brain elastography on a realistic brain geometry obtained from medical imaging.

中文翻译：

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频域毕奥孔隙弹性方程的稳定总压公式：数值分析和应用


这项工作的重点是频域中多孔弹性材料动力学的数值求解。我们基于 Fredholm 替代方案在连续情况下的应用提供了详细的稳定性分析，并考虑了 Biot 方程的总压力公式。在离散设置中，我们提出了一种稳定的等阶有限元方法，辅以额外的压力稳定，以增强数值方案在流体渗透率方面的鲁棒性。利用 Fredholm 替代方案，我们将适定性结果扩展到离散设置，获得线性有限元情况的理论最佳收敛性。我们提出了不同的数值实验来验证所提出的方法。首先，我们考虑具有二维和三维已知解析解的模型问题。接下来，我们证明该方法对于各种渗透率（包括不连续系数的情况）都是稳健的。最后，我们展示了在从医学成像获得的真实大脑几何形状上模拟大脑弹性成像的应用。