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→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 |∇ u|p−2∇ 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.
Related
- Local behavior of solutions to inhomogeneous singular parabolic p-Laplace equations
- Asymptotically Hölder gradient regularity for degenerate or singular parabolic normalized $ p$-Laplace equations
- Gradient Hölder regularity for singular fractional p-Laplace equations
- On sharp singular Moser-Trudinger type inequalities and applications to zero mass (p,Q)-Laplace subelliptic equations in the Heisenberg group
- Gradient rearrangement estimates for singular and degenerate anisotropic p-Laplace equations
BibTeX
@misc{diego2026transform,
title = {{Transform before linearizing: robust Newton methods for singular $p$-Laplace and $p$-Stokes equations}},
author = {Gonzalo G. De Diego and Fortino Garcia and Georg Stadler and Kewei Wang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.12241},
eprint = {2609.12241},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.12241},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Transform before linearizing: robust Newton methods for singular p-Laplace and p-Stokes equations
AU - Gonzalo G. De Diego
AU - Fortino Garcia
AU - Georg Stadler
AU - Kewei Wang
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2609.12241
UR - https://arxiv.org/abs/2609.12241
AB - 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→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 |∇ u|^p−2∇ 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.
ER -