Supraconvergence of a finite difference scheme for solutions in Hs ( 0 , L )

In this paper we study the convergence of a centred finite difference scheme on a non-uniform mesh for a 1D elliptic problem subject to general boundary conditions. On a non-uniform mesh, the scheme is, in general, only first-order consistent. Nevertheless, we prove for s ∈ (1/2, 2] order O(hs)-convergence of solution and gradient if the exact solution is in the Sobolev space H1+s(0, L), i.e. the so-called supraconvergence of the method. It is shown that the scheme is equivalent to a fully discrete linear finite-element method and the obtained convergence order is then a superconvergence result for the gradient. Numerical examples illustrate the performance of the method and support the convergence result.