先变换再线性化：针对奇异 $p$-Laplace 和 $p$-Stokes 方程的鲁棒牛顿方法

中文摘要 AI 辅助

对于 $1<p<2$，$p$-Laplace 方程及其 $p$-Stokes 推广在数值上难以求解。牛顿法仅在接近解时快速收敛，其迭代次数随网格细化而增加，并在 $p\ o1$ 时恶化。更鲁棒的 Picard 迭代仅线性收敛。我们不采用全局化或预条件化牛顿法，而是修改其所应用的系统。我们通过引入通量 $|\ abla u|^{p-2}\ abla u$ 作为辅助（或“提升”）变量来提升方程，对所得本构关系施加非线性变换，进行线性化，并通过静态凝聚消除辅助变量。仅提升不会改变线性化；正是先行的变换产生了新方法。消除是代数性的且在求积点逐点进行，因此通量变量从未被离散化，不产生 inf-sup 条件或不定系统，每次迭代的成本与标准牛顿步相同。结合对提升变量的逐点可行性界（该界保持扩散张量一致正定），我们证明该迭代在有限维情形下对 $p$-Laplace 方程全局收敛且局部具有二次收敛速率；一维模型问题解释了为何滞后的通量变量消除了标准牛顿法在 $p$ 接近 1 时的锯齿行为。基于 Firedrake 的二维和三维 $p$-Laplace 问题以及稳态和瞬态 $p$-Stokes 流动实验表明，迭代次数对 $p$ 和网格细化基本不敏感，并且在 $p$ 接近 1 时比标准牛顿法少多达一个数量级的迭代次数。

英文摘要

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 $p\to1$. 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.

发表机构

• Courant Institute School of Mathematics, Computing and Data Science, New York University（纽约大学柯朗数学科学研究所）
• ETSIT, Technical University of Madrid (UPM)（马德里理工大学ETSIT）

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