It is well known that via the augmented Lagrangian method, one can solve Stokes' system by solving the nearly incompressible linear elasticity equation. In this paper, we show that the converse holds, and approximate the inverse of the linear elasticity operator with a convex linear combination of parameter-free operators. In such a way, we construct a uniform preconditioner for linear elasticity for all values of the Lam\'{e} parameter $\lambda\in [0,\infty)$. Numerical results confirm that by using inf-sup stable finite-element spaces for the solution of Stokes' equations, the proposed preconditioner is robust in $\lambda$.

翻译：本文研究了通过增广拉格朗日方法可以通过求解几乎不可压的线性弹性方程组解决斯托克斯方程的问题。同时，本文还证明了逆线性弹性算子可以通过使用无需参数的算子的凸线性组合来近似。在这样的前提下，我们为线性弹性构造了一个统一的预条件化器，适用于 $\lambda \in [0, \infty)$ 的所有值。数值实验证实了在使用基于 inf-sup 稳定的有限元空间求解斯托克斯方程时，所提出的预条件化方法在 $\lambda$ 值上具有鲁棒性。