We study the convergence of a finite difference scheme that approximates the third initial-boundary-value problem for parabolic equation with variable coefficients on a unit square. We assume that the generalized solution of the problem belongs to the Sobolev space $W_2^{s,s/2}$, $\,s\leq 3$. An almost second-order convergence rate estimate (with additional logarithmic factor) in the discrete $W^{1,1/2}_2$ norm is obtained. The result is based on some nonstandard a priori estimates involving fractional order discrete Sobolev norms.