Potential flow is computed around multi-element airfoils in two dimensions using numerical conformal mapping combined with the calculus of the Schottky-Klein prime function. The corners at the trailing edges of the airfoils are successively removed by Kármán–Trefftz maps. The map from the domain exterior to disks to the domain exterior to the smooth images of the airfoils is computed using an extension of a Fourier series method for the disk due to Fornberg. The velocity potential for flow in the circle domain, with circulation calculated to satisfy the Kutta condition in the airfoil domain, is computed using the Schottky-Klein prime function. The computations are compared to examples from the aerospace engineering literature.