We present a new method for investigating the transport of an active chemical component in a convective flow. We apply a three‐dimensional front tracking method using a triangular mesh. For the refinement of the mesh we use subdivision surfaces which have been developed over the last decade primarily in the field of computer graphics. We present two different subdivision schemes and discuss their applicability to problems related to fluid dynamics. For adaptive refinement we propose a weight function based on the length of triangle edge and the sum of the angles of the triangle formed with neighboring triangles. In order to remove excess triangles we apply an adaptive surface simplification method based on quadric error metrics. We test these schemes by advecting a blob of passive material in a steady state flow in which the total volume is well preserved over a long time. Since for time‐dependent flows the number of triangles may increase exponentially in time we propose the use of a subdivision scheme with diffusive properties in order to remove the small scale features of the chemical field. By doing so we are able to follow the evolution of a heavy chemical component in a vigorously convecting field. This calculation is aimed at the fate of a heavy layer at the Earth's core‐mantle boundary. Since the viscosity variation with temperature is of key importance we also present a calculation with a strongly temperature‐dependent viscosity.