On the convergence of finite-difference schemes for parabolic equations with variable coefficients (Q1118365)

For a two-dimensional linear second-order parabolic equation with variable coefficients the solution is supposed to be belonging to the Sobolev space \(W_ 2^{\lambda,\lambda /2}\) \((1<\lambda \leq 3)\). For its solution an implicit finite difference scheme is used which satisfies the condition \(c_ 1h^ 2\leq \tau \leq c_ 2h^ 2\). Stability and convergence in \(W_ 2^{1,1/2}\) norm are proved, and the estimate of convergence is compatible with the smoothness of the analytical solution. For the proof a bilinear version of the Bramble-Hilbert lemma is used.