A boundary element method for analysing thermocapillary convection with a free surface has been developed. The divergence theorem is applied to the non-linear convective volume integral in the boundary formulation with the pressure penalty function. Consequently, velocity and temperature gradients are eliminated and the complete formulation is written in terms of velocity and temperature. This provides considerable reduction in storage and computational requirements while improving accuracy. Employing this method, a simulation of surface tension driven convective flow in a rectangular cavity with differential heated isothermal lateral walls in a microgravity environment has been demonstrated. The of different Marangoni numbers, Reynolds numbers and Prandtl numbers on the shape of the free surface, the temperature distribution and the flow fields has also been studied.