We consider the Dirichlet problem on a strip for a singularly perturbed elliptic equation of reaction-diffusion type.For such a problem well-known finite difference schemes converge $\eps$-uniformly with the order of accuracy not higher than second. This can imply some restrictions for practical use of these schemes in applications. Base on special schemes on {\it piecewise uniform} meshes and by using the Richardson extrapolation technique on a sequence of {\it embedded} meshes, we construct a difference scheme that converges $\eps$-uniformly with third-order accuracy up to a logarithmic factor and with the fourth-order accuracy with respect to the variables which are respectively orthogonal and tangent to a boundary. For the above Richardson scheme we construct a decomposition method on overlapping subdomains for which $\eps$-uniform accuracy of the Richardson scheme is preserved.