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：众所周知，通过增广拉格朗日方法，可以借助求解近不可压缩线性弹性方程来求解斯托克斯系统。本文证明了相反结论亦成立，即通过凸线性组合无参数算子来近似线性弹性算子的逆。由此，我们为所有拉梅参数$\lambda\in [0,\infty)$值下的线性弹性问题构建了一个统一的预条件子。数值结果证实，采用满足inf-sup稳定条件的有限元空间求解斯托克斯方程时，所提出的预条件子对$\lambda$具有鲁棒性。