Investigation of a two-grid method of improved accuracy for the elliptic reaction–diffusion equation with boundary layers

A two-grid method for the elliptic equation with a small parameter $\varepsilon$ multiplying the highest derivative is investigated. The $\varepsilon$-uniformly convergent difference scheme on the Shishkin mesh is considered. To resolve the difference scheme, a two-grid method with $\varepsilon$-uniform interpolation formula is used. To increase the accuracy of the scheme, the Richardson extrapolation in the two-grid method is applied. The results of numerical experiments are discussed. Various iterative methods for implementation of the two-grid algorithm are suggested.