Method N. M. Korobova approximate solution of the Dirichlet problem

The paper discusses the generalization of the method embodiments N. M. Korobov approximate solution of the Dirichlet problem for equations of the form
$$Q\left(\frac{\partial }{\partial x_1},\ldots,\frac{\partial }{\partial x_s}\right)u(\mathbf{x})=f(\mathbf{x}),$$
where the functions $u(\mathbf{x}),f(\mathbf{x}),\varphi(\mathbf{x})$ belongs to the class of functions $E_s^\alpha$ in case of using generalized Parallelepipedal nets $M(\Lambda)$ integral lattices $\Lambda$.
Particular attention is paid to the class of differential operators, consisting of operators $Q\left(\frac{\partial }{\partial x_1},\ldots,\frac{\partial }{\partial x_s}\right)$ with zero kernel. The importance of this class of operators due to the fact that up to a constant solution of differential equations with partial derivatives for these operators is uniquely determined. An example of such an operator is the Laplace operator.
In the work, an approximate solution of the Dirichlet problem for partial differential equations using arbitrary generalized parallelepiped mesh $M(\Lambda)$ integer lattice $\Lambda$ for a certain class of periodic functions and shown that by using an infinite sequence of nested grids is generalized parallelepipedal nets sufficiently fast convergence of the approximate solutions to the function $u(\mathbf{x})$.
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