A variational-difference method for solving the Dirichlet’s problem for pseudodifferential elliptic equation of arbitrary order

In this paper we have obtained a variational-difference scheme, which solves the Dirichlet’s problem for $Au = f$, equations, where $A$ is a pseudodifferential operator according to a symbol $a(\xi)$, which satisfies the following condition: $c_1(1+|\xi|)^{\ast}\leq | a(\xi) \leq c_2(1+|\xi|)^{\ast}$. It has been proved that for the considered scheme the convergence speed order in the $\mathrm{H}_p(\Omega)$ space is equal to 1, and in the $L_2(\Omega)$ space it is $p+1$. The matrix of the obtained algebraic eqyation has a shape of a band with $2p+1$ width.