While solving the elliptic equations in the canonical domain with the Dirichlet boundary conditions by the grid method, it is obviously, that boundary conditions are satisfied precisely. Therefore it is necessary to expect, that close to the domain boundary the accuracy of the corresponding difference scheme should be higher, than in the middle of the domain. The quantitative estimate of this boundary effect first was announced without proves in 1989 in the Reports of the Bulgarian Academy of sciences by the first author. There accuracy of the difference schemes for two-dimensional elliptic equation with variable coefficients in the divergent form has been investigated. In this paper 'weight' a priori estimates, taking into account boundary effect, for traditional difference schemes, which approximate, with the second order, first boundary problem for quasi-linear elliptic type equation, which main part has a not divergent form, have been obtained. The paper ends with numerical experiments, which testify to unim-provement, by the order, of the received 'weight' estimates.
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