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We study the convergence of a finite difference scheme that approximates the  third initial-boundary-value problem for a parabolic equation with variable  coefficients on a unit square. We assume that the generalized solution of the  problem belongs to the Sobolev space W s,s/2 2, s≤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.  [Projekat Ministarstva nauke Republike Srbije, br. 174015]

Finite difference approximation of a parabolic problem with variable coefficients