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

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