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.

翻译：在本篇文章中,首先提出和分析一种薄弱的Galerkin方法,用于处理非单体型的准线性椭圆问题。通过使用布鲁韦尔的固定点技术,得出了能源类规范与2美元标准的工作组解决办法和误差估计数。然后引入了一种有效的双格WG方法,以提高计算效率。双格WG方法的趋同错误在能源类规范中分析。提出了数字实验,以核实我们的理论结论。