Finite difference schemes on nonuniform meshes for parabolic problems with generalized solutions (Q2913970)

The authors prove the convergence of finite difference schemes (FDSs) in discrete \(L^2\)-norm assuming that the generalized solution of the considered initial boundary value problem (IBVP) belongs to the corresponding Sobolev space. The obtained convergence rate estimates are consistent with the smoothness of data. These estimates are obtained by direct estimation of the truncation error, using interpolation theory of function spaces. Thus, the application of the Bramble-Hilbert lemma, which usually involves unnecessary restrictions on the mesh step sizes is avoided. Contrary to FDSs considered, where the right-hand sides of the equations are taken in some intermediate non-mesh points, they replace the right-hand sides with some averaged values. This is necessary because in the problems with generalized solutions, the right-hand sides of the equations may be discontinuous functions.