A finite–element method is developed for determining the critical sliding speed for thermoelastic instability of an axisymmetric clutch or brake. Linear perturbations on the constant–speed solution are sought that vary sinusoidally in the circumferential direction and grow exponentially in time. These factors cancel in the governing thermoelastic and heat–conduction equations, leading to a linear eigenvalue problem on the two–dimensional cross–sectional domain for the exponential growth rate for each Fourier wavenumber. The imaginary part of this growth rate corresponds to a migration of the perturbation in the circumferential direction. The algorithm is tested against an analytical solution for a layer sliding between two half–planes and gives excellent agreement, for both the critical speed and the migration speed. Criteria are developed to determine the mesh refinement required to give an adequate discrete description of the thermal boundary layer adjacent to the sliding interface. The method is then used to determine the unstable mode and critical speed in geometries approximating current multi–disc clutch practice.