Volume 57, Number 3, May-June 2023
|Page(s)||1297 - 1322|
|Published online||12 May 2023|
Stability and discretization error analysis for the Cahn–Hilliard system via relative energy estimates
Numerical Mathematics, Department of Mathematics, Johannes Gutenberg University Mainz, Staudingerweg. 9, 55099 Mainz, Germany
2 Johannes Kepler University Linz and Johann Radon Institute for Compuational and Applied Mathematics, Altenbergerstr. 69, 4040 Linz, Austria
3 Johannes Kepler University Linz, Altenbergerstr. 69, 4040 Linz, Austria
* Corresponding author: firstname.lastname@example.org
Accepted: 14 February 2023
The stability of solutions to the Cahn–Hilliard equation with concentration dependent mobility with respect to perturbations is studied by means of relative energy estimates. As a by-product of this analysis, a weak-strong uniqueness principle is derived on the continuous level under realistic regularity assumptions on strong solutions. The stability estimates are further inherited almost verbatim by appropriate Galerkin approximations in space and time. This allows to derive sharp bounds for the discretization error in terms of certain projection errors and to establish order-optimal a priori error estimates for semi- and fully discrete approximation schemes. Numerical tests are presented for illustration of the theoretical results.
Mathematics Subject Classification: 35A15 / 35K52 / 35K55 / 65M12 / 65M15 / 65M60
Key words: A priori error analysis / nonlinear stability analysis / optimal error estimates / variational methods / Cahn–Hilliard
© The authors. Published by EDP Sciences, SMAI 2023
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