The problem of finding local and volume averaged stresses in a two-dimensional heterogeneous solid is formulated in terms of fundamental point load solutions (Green's function) leading to singular integral equations. The resulting equations are solved approximately using a subdomain method in which closed form solutions for a rectangular subdomain are obtained and utilized to find the full field solution. Previously, closed form solutions for a rectangular subvolume had been found, but only for the case of an assumed constant strain. In the present paper the solution is obtained for a quadratic form which includes not only the usual constant term but also linear and quadratic terms. The advantages of using the higher order solutions is illustrated by finding the local field in a periodic composite with square fibers, The numerical solution takes less than 90 CPU s on a workstation. The solution yields average properties independent of the reference modulus as would be expected for an accurate solution of the singular integral equation and the effective transverse modulus vs volume fraction is close to that from Christensen's model developed for round fibers.