Some Uzawa methods for steady incompressible Navier–Stokes equations discretized by mixed element methods

Improved convergence of the Arrow-Hurwicz iteration for the Navier-Stokes equation via grad-div stabilization and Anderson acceleration

“…with ρ and α being user determined parameters. We note that if ρ = ν −1 then AH is exactly the modified Uzawa algorithm from [8]. This iteration is interesting because it is efficient since the two equations decouple, with the second equation being simple and the first equation requiring a typical convection diffusion solver where one controls the diffusion coefficient (in each iteration) with ρ.…”

An Uzawa iterative method for the natural convection problem based on mixed finite element method

“…In this article, in order to conquer the numerical difficulties mentioned earlier and find an efficient and accurate approximation of the natural convection model, the authors are going to design an Uzawa iterative method combined with a mixed finite element method for this problem, where a decoupling discrete system is solved and no saddle point system is required to solve at each iterative step. Moreover, by developing some techniques and using some ideas in Nochetto and Pyo 30 and Chen et al 36 technically, the authors establish convergence result of the method and find an interval of relaxation parameter, which is crucial for the convergence of the presented method.…”

“…Note that all the above works mainly concerned are linear problems. When it comes to nonlinear problems, Chen et al 36 have constructed some Uzawa‐type iterative methods for solving the steady incompressible Navier–Stokes equations and have proved that the methods converge geometrically with a contraction number. Further, the steady magnetohydrodynamic equations 37 are solved by applying some Uzawa‐type iterative algorithms, which correct pressure at each iteration.…”

The Generalized Arrow-Hurwicz Method with Applications to Fluid Computation

“…More recently, it was proved in [6] that when the method was used for solving incompressible Navier-Stokes equations discretized by mixed element methods, it converges with a contraction number independent of the finite element mesh size h, even for regular triangulations. We remark that the modified Uzawa-type iterative method devised in [5] can be viewed as a specific case of the Arrow-Hurwicz discussed in [6]. Here, motivated by some ideas in [6], we will design a generalized Arrow-Hurwicz method for solving problem P and then establish the corresponding convergence rate analysis.…”