Numerical algorithms based on Galerkin methods for the modeling of reactive interfaces in photoelectrochemical solar cells

Large-scale utilization of photovoltaic (PV) devices, or solar cells, has been hampered for years due to high costs and lack of energy storage mechanisms. Photoelectrochemical solar cells (PECs) are an attractive alternative to conventional solid state PV devices because they are able to directly convert solar energy into hydrogen fuel. The hydrogen fuel can then be used at a later time to generate electricity. Photoelectrochemical solar cells are able to produce fuel through chemical reactions at the interface of a semiconductor and electrolyte when the device is illuminated. In this dissertation, we focus on the modeling and numerical simulation of charge transport in both the semiconductor and electrolyte region as well as their interaction through a reactive interface using the drift-diffusion-Poisson equations.

The main challenges in constructing a numerical algorithm that produces reliable simulations of PECs are due to the highly nonlinear nature of the semiconductor and electrolyte systems as well as the nonlinear coupling between the two systems at the interface. In addition, the evolution problem under consideration is effectively multi-scale in the sense that the evolution of the system in the semiconductor and the corresponding one in the electrolyte evolve at different time scales due to the quantitative scaling differences in their relevant physical parameters. Furthermore, regions of stiffness caused by boundary layer formation where sharp transitions in densities and electric potential occur near the interface and pose severe constraints on the choice of discretization strategy in order to maintain numerical stability.

In this thesis we propose, implement and analyze novel numerical algorithms for the simulation of photoelectrochemical solar cells. Spatial discetizations of the drift-diffusion-Poisson equations are based on mixed finite element methods and local discontinuous Galerkin methods. To alleviate the stiffness of the equations we develop and analyze Schwarz domain decomposition methods in conjunction with implicit-explicit (IMEX) time stepping routines. We analyze the numerical methods and prove their convergence under mesh refinement. Finally, we present results from numerical experiments in order to develop a strategy for optimizing solar cell design at the nano-scale.