Globalization strategies for Newton–Krylov methods for stabilized FEM discretization of Navier–Stokes equations

In this work we study the numerical solution of nonlinear systems arising from stabilized FEM discretizations of Navier–Stokes equations. This is a very challenging problem and often inexact Newton solvers present severe difficulties to converge. Then, they must be wrapped into a globalization strategy. We consider the classical backtracking procedure, a subspace trust-region strategy and an hybrid approach. This latter strategy is proposed with the aim of improve the robustness of backtracking and it is obtained combining the backtracking procedure and the elliptical subspace trust-region strategy. Under standard assumptions, we prove global and fast convergence of the inexact Newton methods embedded in this new strategy as well as in the subspace trust-region procedure. Computational results on classical CFD benchmarks are performed. Comparisons among the classical backtracking strategy, the elliptical subspace trust-region approach and the hybrid procedure are presented. Our numerical experiments show the effectiveness of the proposed hybrid technique.