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 $\tau+h^2$, $\tau^{1/2}+h$, $\tau+h$, depending on the smoothness of the data.