Solving steady incompressible Navier–Stokes equations by the Arrow–Hurwicz method

The Generalized Arrow-Hurwicz Method with Applications to Fluid Computation

“…In addition, since the bilinear form b(•,•) satisfies the condition (2.2), or equivalently, 6) we easily know the image of B T , i.e. Im(B T ), is closed.…”

“…Later on, this method was mentioned in [20] for solving steady incompressible Navier-Stokes equations. 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].…”

“…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.…”

“…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. As shown in [6] that one of the main advantages of the method is that it requires to solve no linear saddle-point systems at each iteration step (except for the initial step).…”

“…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. As shown in [6] that one of the main advantages of the method is that it requires to solve no linear saddle-point systems at each iteration step (except for the initial step). To show the power of our abstract framework, we will apply it to study three specific nonlinear saddle-point problems, including mathematical model and its numerical method for the steady incompressible magnetohydrodynamic equations, the variational multi-scale (VMS) method and the defect-correction method for numerically solving the steady incompressible Navier-Stokes equations.…”