Error estimates for full discretization by an almost mass conservation technique for Cahn--Hilliard systems with dynamic boundary conditions
Provides rigorous error analysis for a class of complex phase-field models, benefiting numerical analysts and computational scientists working on multiphase problems with dynamic boundaries.
The paper proves optimal-order error estimates for full discretization of bulk-surface Cahn-Hilliard systems with dynamic boundary conditions, using linear finite elements and linearly implicit BDF methods up to order five. The approach exploits almost mass conservation to derive a Poincaré-type inequality, and is generalized to Cahn-Hilliard on evolving surfaces.
A proof of optimal-order error estimates is given for the full discretization of the bulk--surface Cahn--Hilliard system with dynamic boundary conditions in a smooth domain. The numerical method combines a linear bulk--surface finite element discretization in space and linearly implicit backward difference formulae of order one to five in time. The error estimates are obtained by a consistency and stability analysis, based on an energy estimate and the novel approach of exploiting the almost mass conservation of the error equations to derive a Poincaré-type inequality. We demonstrate how this approach can be generalized to other almost mass conserving problems. To this end we prove optimal-order fully discrete error estimates for the Cahn--Hilliard equation on evolving surfaces. We illustrate and complement our findings by numerical experiments.