  We describe a method of asymptotic approximations to solutions of mixed
boundary value problems for the Laplacian in a three-dimensional domain with
many perforations of arbitrary shape, with the Neumann boundary conditions
being prescribed on the surfaces of small voids. The only assumption made on
the geometry is that the diameter of a void is assumed to be smaller compared
to the distance to the nearest neighbour. The asymptotic approximation,
obtained here, involves a linear combination of dipole fields constructed for
individual voids, with the coefficients, which are determined by solving a
linear algebraic system. We prove the solvability of this system and derive an
estimate for its solution. The energy estimate is obtained for the remainder
term of the asymptotic approximation.
