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

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 lift the equation by introducing the flux $\vert \nabla u\vert^{p−2}\nabla u$ as an auxiliary (or ‘‘lifting’’) variable, apply a nonlinear transformation to the resulting constitutive relation, linearize, and 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.

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