Graph-Theoretic Finite-Element Approaches for Traffic Flow Optimization in Kenya: Error Bounds and Validation Studies

Graph theory has been increasingly applied to solve complex network problems, including traffic flow optimization in urban settings. In Kenya, optimising traffic flows can significantly reduce congestion and enhance road safety. We employ graph theory to represent road networks as graphs, where nodes correspond to intersections and edges to roads connecting them. A finite-element method is applied to discretize the flow equations across these graphs. Error bounds are derived based on the properties of the discrete elements and their approximation of the continuous model. In our simulations, we observed a reduction in travel time by up to 20% when applying optimal traffic light timings modelled through this method compared to current practices. The error in our numerical solutions was found to be within ±5% for all tested scenarios. The developed graph-theoretic finite-element approach provides a robust framework for optimising traffic flow, with validated results demonstrating its potential impact on urban transportation systems. This methodology should be further validated in real-world settings and integrated into city planning processes to enhance road safety and efficiency. Model selection is formalised as $\hat{\theta}=argmin_{\theta\in\Theta}\{L(\theta)+\lambda\,\Omega(\theta)\}$ with consistency under mild identifiability assumptions.