Mixed Finite Element Methods for Addressing Multi-Species Diffusion Using the Stefan-Maxwell Equations

Abstract

The Stefan-Maxwell 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 make finding solutions difficult, so numerical methods are often used. In the engineering literature the system is inverted to write fluxes as functions of the species gradient before any numerical method is applied. In this thesis it is shown that employing a mixed finite element method makes the inversion unnecessary, allowing the numerical solution of Stefan-Maxwell equations in their primitive form. The plan of the thesis is as follows, first a mixed variational formulation will be derived for the Stefan-Maxwell equations. The nonlinearity will be dealt with through a linearization. Conditions for well-posedness of the linearized formulation are then determined. Next, the linearized variational formulation is approximated using mixed finite element methods. The finite element methods will then be shown to converge to an approximate solution. A priori error estimates are obtained between the solution to the approximate problem and the exact solution. The convergence order is then verified through an analytic test case and compared to standard methods. Finally, the solution is computed for another test case involving the diffusion of three species and compared to other methods.