Efficient three-level scheme for parabolic equations in cylindrical coordinates in a region with a small hole

An efficient three-level scheme for parabolic equations in cylindrical coordinates is constructed in a region with a small hole. No axial symmetry is assumed. The convergence rate of the scheme is estimated under minimum requirements on the initial data. The estimates are uniform with respect to a small parameter—the inner diameter of the region. The order of convergence is τ + h ², τ^1/2 + h, τ + h, depending on the smoothness of the data.