The proposed Quasi-Newton based scheme can be extended to handle time-dependent or coupled magnetoelectric problems by incorporating a time discretization approach alongside the existing iterative framework. In time-dependent scenarios, the governing equations can be formulated using a time-stepping method, such as the implicit or explicit Euler method, or higher-order Runge-Kutta methods. Each time step would involve solving a nonlinear system similar to the one described in the context, where the local Quasi-Newton updates can be applied to approximate the local permeabilities at each time step.
For coupled magnetoelectric problems, the scheme can be adapted by simultaneously solving the magnetic and electric field equations. This involves defining a coupled system of equations that includes both the magnetic field intensity and the electric field, potentially represented through a vector potential formulation. The Quasi-Newton updates can be utilized to derive local permeabilities that account for the interaction between the magnetic and electric fields, ensuring that the material behavior is accurately captured. Additionally, the iterative solver can be modified to handle the coupling by updating both fields in a staggered or simultaneous manner, depending on the specific problem requirements.