The paper presents a high order compact finite difference scheme constructed on a 19-point stencil using Steklov averaging operators, for the Dirichlet boundary valueproblem for three-dimensional convection-diffusion equation with constant coefficients in the unit cube:
The first part is an introduction concerning Sobolev spaces and appropiate norms, the finite difference quotients (forward, backward, and central, respectively).
The second part focuses on the construction of the derivation of the 19-points compact finite difference schemes for (1).
In the third part the authors derive an a priori estimate of the discretization error, of type:
where is the solution of the finite difference scheme and denotes a positive generic constant, independent of and
Thus, the real parameter satisfies under the assumption that the solution of the original boundary-value problem (1) belongs to the Sobolev space
The error estimates are derived using techniques that employ the generalized Bramble-Hilbert lemma.
The fourth part concerns the estimate of the convergence rate.
The comparisons with other methods and the main conclusions are given in the last part.