The method of total approximation of the solution of the Dirichlet problem for a multidimensional Sobolev-type equation

We study the Dirichlet problem for a multidimensional differential equation of Sobolev type with variable coefficients. The considered equation is reduced to an integro-differential equation of parabolic type with a small parameter. For an approximate solution of the obtained problem, a locally one-dimensional difference scheme is constructed. Using the method of energy inequalities, an a priori estimate is obtained for the solution of a locally one-dimensional difference scheme, which implies its stability and convergence. For a two-dimensional problem, an algorithm for the numerical solution of the problem posed was constructed, numerical experiments were carried out on test examples, illustrating the theoretical results obtained in this work.