A regularized Newton-like method for nonlinear PDE

TL;DR

This paper introduces an adaptive regularization method for Newton-like iterations in finite element solutions of nonlinear PDEs, improving stability and convergence starting from coarse meshes.

Contribution

It develops a novel adaptive regularization strategy combining Tikhonov regularization and pseudo-transient continuation for nonlinear PDEs on coarse meshes.

Findings

Proves local q-linear convergence of the method.

Demonstrates the effectiveness through numerical examples.

Shows the method stabilizes iterations on coarse meshes.

Abstract

An adaptive regularization strategy for stabilizing Newton-like iterations on a coarse mesh is developed in the context of adaptive finite element methods for nonlinear PDE. Existence, uniqueness and approximation properties are known for finite element solutions of quasilinear problems assuming the initial mesh is fine enough. Here, an adaptive method is started on a coarse mesh where the finite element discretization and quadrature error produce a sequence of approximate problems with indefinite and ill-conditioned Jacobians. The methods of Tikhonov regularization and pseudo-transient continuation are related and used to define a regularized iteration using a positive semidefinite penalty term. The regularization matrix is adapted with the mesh refinements and its scaling is adapted with the iterations to find an approximate sequence of coarse mesh solutions leading to an efficient approximation of the PDE solution. Local q-linear convergence is shown for the error and the residual in the asymptotic regime and numerical examples of a model problem illustrate distinct phases of the solution process and support the convergence theory.