A weak Galerkin method and its two-grid algorithm for the quasi-linear elliptic problems of non-monotone type

TL;DR

This paper introduces a weak Galerkin method for non-monotone quasi-linear elliptic problems, providing existence, error estimates, and an efficient two-grid algorithm with verified numerical results.

Contribution

The paper develops a novel weak Galerkin approach and a two-grid algorithm for complex elliptic problems, enhancing computational efficiency and theoretical understanding.

Findings

Existence of WG solution established using Brouwer's fixed point theorem.

Error estimates derived in energy-like and L2 norms.

Numerical experiments confirm theoretical results.

Abstract

In this article, a weak Galerkin method is firstly presented and analyzed for the quasi-linear elliptic problem of non-monotone type. By using Brouwer's fixed point technique, the existence of WG solution and error estimates in both the energy-like norm and the $L^2$ norm are derived. Then an efficient two-grid WG method is introduced to improve the computational efficiency. The convergence error of the two-grid WG method is analyzed in the energy-like norm. Numerical experiments are presented to verify our theoretical findings.