A locally one-dimensional scheme for the third initial boundary value problem for a multidimensional Sobolev-type equation with a memory effect

The multidimensional Sobolev equation with memory effect and boundary conditions of the third kind is studied. For the numerical solution of the problem, the original multidimensional problem is reduced to the third initial boundary value problem for an integro-differential equation of parabolic type with a small parameter. The convergence of the solution of the obtained modified problem to the solution of the original problem when the small parameter approaches zero is proved. A. A. Samarsky's local one-dimensional difference scheme is used for the modified problem, the main idea of which is to reduce the transition from layer to layer to the sequential solution of a number of one-dimensional problems in each of the coordinate directions. In this case, the approximation error of the additive scheme is defined as the sum of inconsistencies for all intermediate schemes, that is, the constructed additive scheme has a total approximation, so that each of the intermediate schemes of the chain may not approximate the original problem, the approximation is achieved by summing up all inconsistencies for all intermediate schemes. Using the method of energy inequalities, a priori estimates are obtained, from which the uniqueness and stability of the solution of the locally one-dimensional difference scheme, as well as the convergence of the solution of the scheme to the solution of the original differential problem, follow.