Implicit and efficient schemes for a parabolic equation in a spherical layer

An implicit and an efficient three-level scheme for a parabolic equation in spherical coordinates is constructed in a spherical layer. No axial symmetry is assumed. The convergence rates of the schemes are estimated under minimum requirements on the initial data. The estimates are uniform with respect to the inner diameter of the domain. The order of convergence is $\tau^{\alpha/2}+h^\alpha$, $\alpha=1,2$, depending on the smoothness of the data. The results remain valid for a domain without a hole.