AbstractWe derive defect correction scheme for constructing the sequence of
polynomials of best approximation in the uniform norm to 1/x on a
finite interval with positive endpoints. As an application, we
consider two-level methods for scalar elliptic partial differential
equation (PDE), where the relaxation on the fine grid uses the
aforementioned polynomial of best approximation. Based on a new
smoothing property of this polynomial smoother that we prove,
combined with a proper choice of the coarse space, we obtain as a
corollary, that the convergence rate of the resulting two-level
method is uniform with respect to the mesh parameters, coarsening
ratio and PDE coefficient variation.