The Chebyshev-Jacobi smoother, while effective in the matrix-free context, may have limitations when dealing with highly anisotropic or heterogeneous systems. One potential limitation is the uniformity of the smoothing operation across all directions, which may not be optimal for systems with varying material properties or complex geometries. In such cases, the Chebyshev-Jacobi smoother may struggle to adequately address the anisotropy or heterogeneity present in the system, leading to suboptimal convergence rates or accuracy.
To overcome these limitations, alternative smoothing techniques can be explored in the matrix-free context. One approach is to implement more sophisticated smoothers, such as Gauss-Seidel or symmetric successive over-relaxation (SSOR), that can adapt to the local characteristics of the system. These smoothers can be tailored to address anisotropy by considering directional variations in the material properties and adjusting the smoothing operation accordingly.
Moreover, techniques like algebraic multigrid methods or hybrid smoothers that combine different relaxation schemes can offer improved convergence properties for challenging systems. By exploring a diverse range of smoothing techniques and customizing them to suit the specific characteristics of the phase-field fracture model, the matrix-free solver can overcome the limitations of the Chebyshev-Jacobi smoother and enhance its performance in handling complex scenarios.