Yes, the local Quasi-Newton updates can be combined with other iterative solvers beyond the fixed-point and Newton-type methods. For instance, they can be integrated with optimization-based iterative methods, such as the Gauss-Seidel or Jacobi methods, which are commonly used for solving linear systems. By incorporating local Quasi-Newton updates into these frameworks, one can enhance the convergence properties and efficiency of the solvers, particularly in cases where the underlying problem exhibits strong nonlinearity.
Additionally, the local Quasi-Newton updates can be utilized in conjunction with advanced techniques such as the Broyden's method or other quasi-Newton methods that focus on updating the Jacobian approximation in a more generalized manner. This flexibility allows for the development of hybrid iterative schemes that leverage the strengths of multiple approaches, potentially leading to improved convergence rates and robustness in solving complex nonlinear problems.
Moreover, the integration of local Quasi-Newton updates with multigrid methods could also be explored, where the updates are applied at different levels of mesh refinement to accelerate convergence across scales. This combination could be particularly beneficial in large-scale simulations where computational efficiency is critical.