The paper considers grad-div stabilized equal-order finite elements (FE) methods for the linearized Navier-Stokes equations. A block triangular preconditioner for the resulting system of algebraic equations is proposed which is closely related to the Augmented Lagrangian (AL) preconditioner. A field-of-values analysis of a preconditioned Krylov subspace method shows convergence bounds that are independent of the mesh parameter variation. Numerical studies support the theory and demonstrate the robustness of the approach also with respect to the viscosity parameter variation, as is typical for AL preconditioners when applied to inf-sup stable FE pairs. The numerical experiments also address the accuracy of grad-div stabilized equal-order FE method for the steady state Navier-Stokes equations.

翻译：本文研究线性化 Navier-Stokes 方程的 grad-div 稳定化等阶有限元方法。针对由此产生的代数方程组，提出了一种与增广拉格朗日预条件子密切相关的块三角预条件子。对预条件 Krylov 子空间方法进行的数值域分析表明，其收敛界与网格参数的变化无关。数值研究支持了该理论，并证明了该方法对于粘度参数变化同样具有鲁棒性，这与增广拉格朗日预条件子应用于 inf-sup 稳定有限元对时的典型表现一致。数值实验还探讨了 grad-div 稳定化等阶有限元方法对于稳态 Navier-Stokes 方程的精度。