  We consider a boundary value problem in unbounded 2D doubly periodic
composite with circular inclusions having arbitrary constant conductivities. By
introducing complex potentials, the boundary value problem for the Laplace
equation is transformed to a special R-linear BVP for doubly periodic analytic
functions. This problem is solved with use of the method of functional
equations. The R-linear BVP is transformed to a system of functional equations.
A new improved algorithm for solution of the system is proposed. It allows one
not only to compute the average property but to reconstruct the solution
components (temperature and flux) at an arbitrary point of the composite.
Several computational examples are discussed in details demonstrating high
efficiency of the method. Indirect estimate of the algorithm accuracy has been
also provided.
