Long time H1-norm stability and suboptimal convergence of a fully discrete L2-version compact difference method on nonuniform mesh grids to solve fourth-order subdiffusion equations

This study introduces a designed L2 compact approach for the fourth-order subdiffusion models on general nonuniform meshes. Initially, we establish long-time $H^1$-stability and estimation of the error for the L2 compact approach under general nonuniform meshes, imposing only mild conditions on the time step ratio ${\rho }_k$. This is achieved by leveraging the positive semidefiniteness of an essential bilinear form linked to the operator of L2 fractional derivative discussed by Quan and Wu (2023) [26]. Subsequently, we proceed to discretize the spatial differentiation using the compact difference approach, resulting in a fully discrete scheme that achieves the convergence of fourth-order for the space variable. Additionally, we demonstrate that the rate of convergence in the $H^1$-norm is the half of $(5-\alpha )$ for the improved graded mesh grids. Lastly, we conduct some numerical experiments to validate the robustness and competitiveness of our suggested approach.