Auxiliary Gradient-Flow Solvers for Generalized Newtonian Models

Abstract

We introduce an auxiliary gradient-flow framework for variational problems with generalized Newtonian structure governed by an N-function. The key idea is to replace the nonlinear constitutive dependence on the gradient, or symmetric gradient, by an auxiliary scalar variable representing its squared magnitude. This shifts the nonlinearity from the state equation to the auxiliary variable, yielding a sequence of uniformly elliptic weighted linear problems. At the continuous level, we construct an auxiliary energy on a metric space adapted to the growth of the underlying N-function. In this topology, we prove lower semicontinuity, geodesic $λ$-convexity, and exponential convergence of the associated minimizing-movement scheme. At the finite element level, we derive a metric gradient flow through an explicit Riesz map, prove global well-posedness of the resulting semi-discrete ODE, and establish convergence to the finite element solution of the Euler--Lagrange equations of the generalized Newtonian energy. For the $p$-Laplacian and $p$-Stokes models, this gives a rigorous convergence result for $4/3\le p\le 4$, $p\ne2$, with asymptotic rate estimates beyond this range. We also propose practical time discretizations, including an operator-splitting scheme that gives the \kac iteration as a special case, and an adaptive pseudo-transient method that can be implemented using scalable linear solvers. Numerical experiments for power-law, Carreau--Yasuda, regularized Bingham, and optimal-design models demonstrate robustness, mesh-independent iteration counts in the tested regimes, and performance that matches or outperforms Newton's method.