Abstract Convection problems governed by hyperbolic equations cannot be solved by conventional finite element methods. Various upwind schemes introduced into finite element methods to solve this class of problems suffer from a numerical error called the artificial diffusion. The amount of artificial diffusion depends not only on the mesh size but also on the flow rate. To minimize artificial diffusion the method of internal nodes is introduced. In this method, which is based on the Lagrangian interpretation of the first order upwind scheme, the flow within each nodal volume can be considered as an one-dimensional flow regardless of the number of upstream or downstream nodes. The use of internal nodes can provide an accurate solution to the degree of cure convection problem on a more coarse mesh with significantly reduced CPU cost. Sample problems are solved to validate the accuracy and efficiency of the method. When incorporated with the heat transfer equations, the use of internal nodes underscores the fact that some RTM problems are very sensitive to errors caused by artificial diffusion.