Summary: We present numerical analysis of boundary value problems arising in plane compressible subsonic flows in a double-connected domain. The conformal mapping of a circular annulus on the domain of the flow and the Newton iterations method permit to reduce the original physical problem to a sequence of nontraditional boundary value problems in a rectangle with the ratio of sides defined by the conformal radius of the physical domain. These boundary value problems are investigated numerically using finite elements of higher order (bicubic elements) and numerical conformal mapping. Special attention is paid to the analysis of a flow around a profile near the surface of the earth.