Convergence of fourth order compact difference schemes for three-dimensional convection-diffusion equations

We consider a Dirichlet boundary-value problem for the three-dimensional convection-diffusion equations with constant coefficients in the unit cube. A high order compact finite difference scheme is constructed on a 19-point stencil using the Steklov averaging operators. We prove that the finite difference scheme converges in discrete W-2(m) (omega)-norm with the convergence rate O(h(s-m)), where the real parameter s satisfies the condition max(1.5, m) < s <= m + 4, m = 0, 1, 2, and the exact solution belongs to the Sobolev space W-2(s) (Omega).