An Adaptive Finite Element Method Based on Superconvergent Cluster Recovery for the Cahn-Hilliard Equation

In this study, we construct an error estimate for a fully discrete finite element scheme that satisfies the criteria of unconditional energy stability, as suggested in [1]. Our theoretical findings, in more detail, demonstrate that this system has second-order accuracy in both space and time. Additionally, we offer a powerful space and time adaptable approach for solving the Cahn-Hilliard problem numerically based on the posterior error estimation. The major goal of this technique is to successfully lower the calculated cost by controlling the mesh size using a Superconvergent Cluster Recovery (SCR) approach in accordance with the error estimation. To demonstrate the effectiveness and stability of the suggested SCR-based algorithm, numerical results are provided.