Iterative Galerkin Discretizations for Strongly Monotone Problems

TL;DR

This paper presents a finite element fixed-point iteration method for strongly monotone quasi-linear elliptic PDEs that leverages monotonicity to reuse the iteration matrix, improving efficiency and convergence.

Contribution

It introduces a novel fixed-point iteration approach exploiting monotonicity, reducing computational effort by avoiding repeated matrix recomputations, with proven error estimates and optimal convergence analysis.

Findings

The method achieves optimal convergence rates.

The number of iterations can be bounded based on mesh size or polynomial degree.

Numerical experiments confirm theoretical results.

Abstract

In this article we investigate a finite element formulation of strongly monotone quasi-linear elliptic PDEs in the context of fixed-point iterations. As opposed to Newton's method, which requires information from the previous iteration in order to linearise the iteration matrix (and thereby to recompute it) in each step, the alternative method used in this article exploits the monotonicity properties of the problem, and only needs the iteration matrix calculated once for all iterations of the fixed-point method. We outline the a priori and a posteriori error estimates for iteratively obtained solutions, and show both theoretically as well as numerically how the number of iterations of the fixed-point method can be restricted in dependence of the mesh size, or of the polynomial degree, to obtain optimal convergence.