On the long-time behavior of the continuous and discrete solutions of a nonlocal Cahn–Hilliard type inpainting model
Dandan Jiang, Mejdi Azaïez, Alain Miranville, Chuanju Xu, Hui Yao · Mathematics and Computers in Simulation · 2024
In this paper, we study the analytical and numerical long-time stability of a nonlocal Cahn–Hilliard model with a fidelity term for image inpainting introduced in our previous work (Jiang et al., 2024). First, we establish the uniform boundedness of the continuous problem in both L 2 and H 1 spaces, which is obtained by using the Gagliardo–Nirenberg inequality and the uniform Grönwall lemma. Then, for the temporal semi-discrete scheme, the uniform estimates in L 2 and H 1 spaces are derived with the aid of the discrete uniform Grönwall lemma under a suitable assumption on the nonlinear potential. This demonstrates the long-time stability of the proposed scheme in L 2 and H 1 spaces. Finally, we validate the long-time stability and the applicability of our method in signal reconstruction and image inpainting. These numerical experiments demonstrate the high effectiveness of our proposed model.