Unlike common approaches this paper does not rely on results for 3D problems. Degeneration of the operator on a part of the border requires the use of the corresponding weight functional spaces to analyze a variational statement to build a difference scheme and to prove the convergence. A result of convergence is proven using piece wise linear finite elements. The estimate of the convergence rate of the approximate solution to the exact one is not worse than in the case of non-degenerate equation. A numerical example that confirms the estimate is given.