Difference approximation methods for problems of mathematical physics

In the paper we construct and investigate conservative schemes for elliptic equations in an arbitrary domain. To obtain difference approximations in the case of equations with mixed derivatives and with boundary conditions of the third kind, the concept of a vector scheme proves to be useful. Vector difference schemes are constructed by means of the integro-interpolation method (balance method).
To obtain economical algorithms for the solution of many-dimensional parabolic problems we use the method of summary approximations, which leads to additive schemes and vector additive schemes. In particular, we construct economical additive vector schemes for parabolic equations with boundary conditions of the third kind in an arbitrary domain.