Finite difference approximation of a parabolic problem with variable coefficients
parabolic initial-boundary-value problem
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 is considered. 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 obtained error bounds are compatible with the smoothness of the data. The result is based on some nonstandard a priori estimates involving fractional order discrete Sobolev norms.
scientific article; zbMATH DE number 6598860