Stabilized and inexact adaptive methods for capturing internal layers in quasilinear PDE

TL;DR

This paper introduces a novel adaptive method for solving quasilinear diffusion problems with internal layers, using stabilized inexact solves and mesh refinement to accurately capture complex solution features.

Contribution

It presents a new framework with closed-form parameters for stabilization and proves convergence of the residual, improving internal layer resolution in quasilinear PDEs.

Findings

Method effectively captures internal layers in numerical experiments.

Stabilization parameters are explicitly defined for each mesh refinement.

Convergence of the residual is theoretically established.

Abstract

A method is developed within an adaptive framework to solve quasilinear diffusion problems with internal and possibly boundary layers starting from a coarse mesh. The solution process is assumed to start on a mesh where the problem is badly resolved, and approximation properties of the exact problem and its corresponding finite element solution do not hold. A sequence of stabilized and inexact partial solves allow the mesh to be refined to capture internal layers while an approximate solution is built eventually leading to an accurate approximation of both the problem and its solution. The innovations in the current work include a closed form definition for the numerical dissipation and inexact scaling parameters on each mesh refinement, as well as a convergence result for the residual of the discrete problem. Numerical experiments demonstrate the method on a range of problems featuring steep internal layers and high solution dependent frequencies of the diffusion coefficients.