On the convergence of Variational multiscale methods based on Newton’s iteration for the incompressible flows

In this paper, the convergence of a general algorithm with θ$\theta$-type stabilization form for the variational multiscale (VMS) method is presented. Meanwhile, explicit-type and implicit-type algorithms with linear convergence and quadratic convergence are derived from the θ$\theta$-type algorithm, respectively. The combination of explicit-type and implicit-type algorithms are applied to adaptive VMS, which shows good efficiency. Finally, some numerical tests are shown to support the convergence analysis.