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 H1-stability and estimation of the error for the L2 compact approach under general nonuniform meshes, imposing only mild conditions on the time step ratio ρk. 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 H1-norm is the half of (5−α) for the improved graded mesh grids. Lastly, we conduct some numerical experiments to validate the robustness and competitiveness of our suggested approach.