The paper deals with the questions of numerical solutions of regular and quasi-singular classes of inhomogeneous elliptic problems by the optimal (with respect to Kolmogorov's diameter) methods. The method allowing us to extend significantly the class of functions entering into the boundary conditions of these problems is given. POSING THE PROBLEMLet us consider a problem <&? = smooth convex boundary:where L is an elliptic operator with sufficiently smooth coefficients a^, a^ = ad k = d/dxp k = 1,2 (repeated indices assume summation from 1 to 2).In the traditional formulation of the problem S)f the functions / and g are chosen from such spaces & and #, respectively, that the solutions of the problems &£ = 0( ,L,/,0) and 0° = <2>(i2,L,0,g) are of the same smoothness. For example, if = //°( ), then a space H 3 / 2 (dQ) is chosen as ^, and the solutions of the problems &£ and 0° belong to Η 2 (Ω). Hereafter the notation H s stands for the Sobolev-Slobodetskii functional spaces W^ (s > 0).When solving the problem 3)f numerically, one naturally tries to select the spaceŝ and & so that the 'worst* (with respect to & and #) problems &£ and 0° be solvable with the errors of the same order of magnitude using accuracy-order optimal methods.Let us elucidate the above statements. Let 3C be a class of the problems 0f By Ε we denote a functional space whose norm is used to judge the error of the approximate solution of the problems pertaining to 3t.Let us setWe shall omit the operator sign, when the context suggests which operator generates these sets.