An optimal a priori error estimate in the maximum norm for the Il’in scheme in 2D

The Il’in scheme is the most famous exponentially fitted finite difference scheme for singularly perturbed boundary value problems. In 1D, Kellogg and Tsan presented a precise error estimate for the scheme from which uniform first order convergence in the discrete maximum norm can be concluded. This estimate is optimal in the sense that one can supply easy examples with smooth data such that the order of uniform convergence of the Il’in scheme is one. In 2D the problem of proving an optimal error estimate remained open. Emel’janov conducted an error analysis giving uniform convergence orders close to one-half. Within the community of singularly perturbed problems it is a legend that this error estimate is sharp. Correcting this mistaken belief it is proven in the present paper that under some conditions the optimal uniform first order convergence of the Il’in scheme in 1D carries over to the two dimensional case. This new result is corroborated by numerical experiments which also shed light on the question in which cases the convergence rate deteriorates.