In this chapter the boundary value problem is considered for an elliptic reaction-diffusion equation in a domain with a piecewise-smooth boundary. In this problem in a neighborhood of the corner points, edges, and smooth parts of the boundary, the derivatives of the problem solution in each direction orthogonal to the boundary grow without boundary as the parameter ε tends to zero. To construct finite difference schemes, classical difference approximations of the corresponding differential operators are used. Sufficient conditions for ε-uniform convergence of the finite difference schemes are derived. The fulfillment of these conditions is ensured by choosing special grids that condense near the boundary and have a consistent distribution of nodes.