Abstract: We target highly nonlinear fluid models with von Mises yield critera that give rise to optimization problems with Hessians exhibiting a problematic (near) null space upon linearization with Newton’s method.
The null space is caused by a projector-type coefficient in the Hessian, which is created by terms in the objective functional that resemble the $L^1$-norm. This occurs, for example, in nonlinear incompressible Stokes flow in Earth’s mantle with plastic yielding rheology, which effectively limits stresses in the mantle by weakening the viscosity depending on the strain rate. Using a standard Newton linearization for such an application is known to produce severe Newton step-length reductions due to backtracking line search and stagnating nonlinear convergence. Additionally, these effects become increasingly prevalent as the mesh is refined.
We analyze issues with the standard Newton linearization in an abstract setting and propose an improved linearization, which can be applied straightforwardly to Stokes flow with yielding and other applications as total variation regularization.
Finally, numerical experiments compare the standard and improved Newton linearizations in practice. When we employ our improved linearization within our inexact Newton-Krylov method, a fast and highly robust nonlinear solver is attained that exhibits mesh-independent convergence and scales to large numbers of cores with high parallel efficiency.