Mixed finite element methods for addressing multi-species diffusion using the Maxwell–Stefan equations

The Maxwell–Stefan equations are a system of nonlinear partial differential equations that describe the diffusion of multiple chemical species in a container. These equations are of particular interest for their applications to biology and chemical engineering. The nonlinearity and coupled nature of the equations involving many variables rule out analytical solutions, so numerical methods are often used. In the literature the system is inverted to write fluxes as functions of the species gradient before any numerical method is applied. In this paper it is shown that employing a mixed finite element method makes the inversion unnecessary, allowing the numerical solution of Maxwell–Stefan equations in their primitive form. A mixed variational formulation is derived in the general n-ary case. A priori error estimates between the finite element and exact solutions are obtained. The order of convergence of the method is then verified and compared with standard methods using a manufactured solution. Finally, the solution is computed for a test case from the literature involving the diffusion of three species and compared to solutions from other methods.