A uniformly convergent difference scheme for the singular perturbation problem of a high order elliptic differential equation (Q1352718)

The problem discussed is the singularly perturbed partial differential equation \( \epsilon^2 \Delta^2 \phi + A\phi = f\) on a rectangle in the plane, with the conditions \(\phi = 0\) and \(\partial_n \phi = 0\) on the boundary. Here, \(A\) is a linear, second-order elliptic operator. The authors introduce a difference scheme in which the weights are chosen so as to give the exact solution when \(f\) and the coefficients of \(A\) are constant. Energy estimates are used to derive error bounds.