On the conditions that the net method for the Laplace equation converges with speed $h^2$

The paper gives a uniform estimate of order $h^2$ of the error in the net method of solving the Dirichlet problem for the Laplace equation under the assumptions that the modulus of continuity of the second derivatives of the boundary values and the modulus of continuity of the curvature of the region's boundary do not exceed a function satisfying the Dini condition. It is shown that with the removal of the Collatz boundary values the constraints on the boundary values cannot be significantly relaxed in terms of the moduli of continuity of the second derivatives, while the constraints on the moduli of continuity of the curvature of the boundary cannot be completely lifted.