In solving a mixed-type (elliptic-hyperbolic) differential equation in an unbounded region, which is elliptic near infinity, some way must be found to transfer the boundary conditions at infinity to a finite artificial boundary in order to keep the discretized problem finite. The common example of this is transonic flow over an airfoil or wing with subsonic freestream. Here we present an approach which is in many ways analogous to the adaptive wind-tunnel wall concept. Iterative revision of a Dirichlet condition on the common or boundary of the near and far fields results in convergence to a far-field solution that matches the discretized near-field solution in potential and normal derivative across the matching boundary. The far-field equation is either a first-order (FO) Prandtl-Glauert, or a second-order (SO) Poisson-type approximation to the transonic equation. A parameter is easily calculated which gives a good estimate of the accuracy of the far-field solution in either case. Two-dimensional results are given showing the success of the method in reproducing the circulation and Cp for a lifting airfoil. Accurate solutions are given using far-field matching boundaries which are much closer to the airfoil than is permissible with Klunker-type far fields based on multipole expansions. The results are shown to be invariant with the location of the vortex representing the far-field circulation. Thus, we significantly reduce computer time by factors of 3 (FO) and 7 (SO) for mesh density and accuracy equivalent to those of a fixed asymptotic far-field representation. Nonlifting FO calculations for a three-dimensional rectangular wing similarly yield accurate results for a much reduced near field, cutting computer time by more than a factor of 2 in an unoptimized case where the minimum boundary size has not yet been established.