The improved convergence constant obtained through the refined error analysis in practical simulations of the Cahn-Hilliard equation can have several significant applications:
Efficient Simulations: The improved convergence constant allows for more accurate numerical solutions with reduced computational cost. This can lead to more efficient simulations of phase separation phenomena described by the Cahn-Hilliard equation.
Parameter Sensitivity Analysis: The refined error estimates can help in understanding the sensitivity of the numerical solution to parameters like the interface width parameter ε. This can be crucial in optimizing the simulation parameters for accurate results.
Model Validation: By having a better understanding of the convergence behavior of the numerical scheme, researchers can validate the model against analytical solutions or experimental data more effectively.
Optimization Algorithms: The improved convergence constant can be utilized in optimization algorithms that rely on accurate numerical solutions of the Cahn-Hilliard equation. This can lead to better optimization results in material science and other fields.