An iterative two-grid method for strongly nonlinear elliptic boundary value problems

TL;DR

This paper introduces an iterative two-grid method for solving strongly nonlinear elliptic boundary value problems, combining coarse nonlinear solves with fine linear solves, and provides the first convergence analysis for such an approach.

Contribution

It presents a novel iterative two-grid algorithm with a rigorous convergence analysis for strongly nonlinear elliptic problems, supported by numerical experiments.

Findings

Convergence of the proposed method is established.

Numerical experiments confirm efficiency.

The method effectively reduces computational complexity.

Abstract

We design and analyze an iterative two-grid algorithm for the finite element discretizations of strongly nonlinear elliptic boundary value problems in this paper. We propose an iterative two-grid algorithm, in which a nonlinear problem is first solved on the coarse space, and then a symmetric positive definite problem is solved on the fine space. The innovation of this paper lies in the establishment of a first convergence analysis, which requires simultaneous estimation of four interconnected error estimates. We also present some numerical experiments to confirm the efficiency of the proposed algorithm.