Railway brake discs convert the kinetic energy of rail vehicles to thermal energy to achieve braking. This thermal energy deteriorates braking performance, and it is therefore necessary to conduct thermal analyses of brake discs. In this work, we build a finite element method (FEM) model in FEniCSx to investigate the influence of contact areas between brake pads and discs on the temperature of brake discs. The weak form of the nonlinear heat transfer equation has been derived, which accounts for conduction, convection, and radiation. Multiple Neumann boundary conditions are applied. Simulation results are validated with experimental results. With this efficient FEM model, more advanced research related to railway brake discs can be conducted, such as investigating the effect of wear and thermal expansion or designing a new geometry for brake pads and discs.