Transform before linearizing: robust Newton methods for singular $p$-Laplace and $p$-Stokes equations

Abstract

For $1<p<2$, the $p$-Laplace equation and its $p$-Stokes generalization are difficult to solve numerically. Newton's method converges rapidly only close to the solution, with iteration counts that grow under mesh refinement and deteriorate as $p\to 1$. The more robust Picard iteration converges only linearly. Rather than globalizing or preconditioning Newton's method, we modify the system to which it is applied. We \emph{lift} the equation by introducing the flux $|\nabla u{|}^{p-2}\nabla u$ as an auxiliary (or ''lifting'') variable, apply a nonlinear \emph{transformation} to the resulting constitutive relation, \emph{linearize}, and \emph{eliminate} the auxiliary variable by static condensation. Lifting alone leaves the linearization unchanged; it is the preceding transformation that yields the new method. The elimination is algebraic and pointwise at the quadrature points, so the flux variable is never discretized, no inf-sup condition or indefinite system arises, and the cost per iteration is that of a standard Newton step. Together with a pointwise feasibility bound on the lifting variable that keeps the diffusion tensor uniformly positive definite, this yields an iteration that we prove, in finite dimensions and for the $p$-Laplace equation, to converge globally and locally at a quadratic rate; a one-dimensional model problem explains why the lagged flux variable removes the zig-zag behavior of standard Newton for $p$ close to one. Firedrake-based experiments for $p$-Laplace problems in two and three dimensions and for stationary and time-dependent $p$-Stokes flows show iteration counts largely insensitive to $p$ and to mesh refinement, and up to an order of magnitude fewer iterations than standard Newton for $p$ close to one.