The elliptic equation under the nonhomogeneous Dirichlet boundary condition in 2D and 3D cases is solved. A rectangular nonuniform partition of a domain and polylinear finite elements are taken. For the interpolant of the exact solution u, a priori error estimates in the W 1,2 -norm of order O(|h| 1+α ), 0 α 1, are proved provided that u possesses a weakened smoothness of order 2 + α in terms of the Nikolskii or Sobolev spaces. In the case of α = 1 they involve the third order mixed derivatives

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... of u only. Next error estimates are in terms of data. An estimate of order O(|h| 1+α ), 0 < α < 1, is established for the right-hand side f of the equation having a generalized smoothness of order α (which can be the case with respect to all the variables, except one only). In particular, for α = 1 2 the case of discontinuous f is covered under fairly broad assumptions on the curves (surfaces) of discontinuities. An error estimate of higher order O(|h| 2 | log |h|| 1 2 ) is proved for the discontinuities lying on the lines (planes) parallel to the coordinate ones only (but situated arbitrarily with respect to the partition). Error estimates of order O(|h| 2 | log |h|| σ ), σ = 1 2 , 3 2 , are derived in the case of f which is not compatible with the boundary function; for u ∈ W 3,2 , this compatibility is necessary. The proofs are based on some propositions from the theory of functions. The corresponding lower error estimates are also included; they justify the sharpness of the estimates without the logarithmic multipliers. Finally, we prove similar results in the case of 2D linear finite elements and a uniform partition.