On the rate of convergence of Yosida approximation for rhe nonlocal Cahn-Hilliard equation

06/22/2023

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by   Piotr Gwiazda, et al.

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It is well-known that one can construct solutions to the nonlocal Cahn-Hilliard equation with singular potentials via Yosida approximation with parameter λ→ 0. The usual method is based on compactness arguments and does not provide any rate of convergence. Here, we fill the gap and we obtain an explicit convergence rate √(λ). The proof is based on the theory of maximal monotone operators and an observation that the nonlocal operator is of Hilbert-Schmidt type. Our estimate can provide convergence result for the Galerkin methods where the parameter λ could be linked to the discretization parameters, yielding appropriate error estimates.