Novel iteration method for gradient-dependent diffusion problems of divergence type

This paper proposes a new iterative method for divergence-type nonlinear diffusion problems with gradient-dependent coefficients to enhance solution efficiency. The classical Picard iteration often exhibits slow convergence or instability when applied to such nonlinear equations, likely due to its inability to improve the regularity of the iterative sequence. To address this limitation, we first design an iterative scheme at the partial differential equation level, which requires solving a linear diffusion problem in non-divergence form at each step. At the discrete level, through careful reformulation and linearization of the conservative nonlinear difference scheme, the method preserves the solution structure inherent to divergence-form discretization while remaining consistent with the continuous iterative framework. Numerical experiments demonstrate that the proposed method performs well across various problems, including flux-limited and strongly nonlinear diffusion: it reduces the number of iterations by more than half in many cases, achieves convergence even when Picard iteration fails, and maintains comparable numerical accuracy. The results confirm the robustness and efficiency of the proposed approach.