To establish the reliability and efficiency of the a posteriori error estimators for the nonlinear eigenvalue problem in the phase-field setting, several key steps can be taken:
Theoretical Analysis:
Conduct a detailed analysis of the error estimators in the context of the nonlinear optimization problem. This involves understanding the relationship between the phase-field function, eigenvalues, and eigenfunctions, and how errors in these components affect the overall optimization process.
Develop mathematical proofs or arguments to show the convergence properties of the error estimators. This may involve utilizing techniques from functional analysis, optimization theory, and numerical analysis to establish the reliability of the estimators.
Numerical Validation:
Perform extensive numerical experiments to validate the error estimators. This involves comparing the estimators with known solutions or benchmarks to assess their accuracy and efficiency.
Conduct sensitivity analyses to understand how variations in the parameters of the problem impact the performance of the error estimators.
Convergence Analysis:
Utilize the results from the theoretical analysis and numerical validation to demonstrate the convergence of the adaptive algorithm. This includes showing that the estimators converge to zero as the algorithm progresses, indicating the efficiency of the error estimation process.
By combining theoretical insights, numerical experiments, and convergence analysis, the reliability and efficiency of the a posteriori error estimators for the nonlinear eigenvalue problem in the phase-field setting can be established.