Projection difference scheme for a parabolic functional differential equation with two-dimensional transformation of arguments

A nonlinear parabolic functional differential equation with the functional part containing a generalized superposition of the unknown solution and a transformation of the two-dimensional spatial argument is considered. A projection difference scheme for the approximation of the initial Dirichlet boundary value problem in a rectangle is proposed for a wide class of measurable, including noninvertible, transformations. An estimate of the rate of convergence to the generalized solutions of the initial problem of order $O(\tau^{1/4-\gamma}+h^{1/2-2\gamma})$ in the norm $L_2(Q)$ without a priori assumptions on the invertibility of the transformation and without any mesh size matching is obtained.