Numerical methods for multiphysics, multiphase, and multicomponent models for fuel cells

In this dissertation, we design and analyze efficient numerical methods for obtaining accurate solutions to model problems arising in fuel cells. A basic fuel cell model consists of five principles of conservation, namely, mass, momentum, species, charges (electrons and ions), and thermal energy. Overall, transport equations couple with electrochemical processes through source terms to describe reaction kinetics and electro-osmotic drag in the polymer electrolyte.;To model multiphase species transport in the porous media and the gas channel of fuel cells, we consider a multiphase mixture model framework. The diffusivity of the two-phase mixture water conservation equation in this model is nonlinear, discontinuous, and degenerate. To handle this difficulty, we developed efficient and fast nonlinear iterative solvers based on the Kirchhoff transformation and nonlinear Dirichlet-Neumann domain decomposition methods.;To model the coupling between the multiphase flow in the porous media and the viscous flow in the gas channel of fuel cells, we consider the Darcy-Stokes-Brinkman model, which treats both the Darcy equation and the Stokes equation in a single form of partial differential equation (PDE) but with strongly discontinuous viscosity and permeability coefficients. For this model, we develop robust finite element methods that are uniformly stable with respect to the highly discontinuous coefficients and their jumps.;Finally, we develop new numerical methods for the full steady-state 3D multi-physics simulation of liquid-feed direct methanol fuel cells (DMFC), consisting of five fundamental conservation equations: mass, momentum, species, charges, and thermal energy. Fast convergence of nonlinear iteration is achieved in our method.