The Total Approximation Method for the Dirichlet Problem for Multidimensional Sobolev-Type Equations

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