Abstract

In this paper, we present a rigorous optimal error analysis for a semi-discrete finite element method of the coupled Cahn-Hilliard-Navier-Stokes-Darcy model. The optimal convergence order for the phase variable in the L∞(0,T;H1) norm and the velocity in the L∞(0,T;L2) norm are proved with two key techniques. One key technique is establishing the discrete L2(0,T;H2)∩L2(0,T;L∞) boundedness of the numerical solution for the phase variable to handle the nonlinear regional coupling terms. Another key technique adopts the inverse inequality and Ritz projection to bound the L∞ norm of the error of the phase variable in the nonlinear terms that depend on the viscosity and the mobility. This is the first convergence analysis for the sophisticated Cahn-Hilliard-Navier-Stokes-Darcy system. Numerical experiments are also presented to verify the theory and validate the spatial semi-discretization finite element scheme.

Department(s)

Mathematics and Statistics

Publication Status

Full Text Access

Comments

National Science Foundation, Grant 15303224

Keywords and Phrases

Cahn-Hilliard-Navier-Stokes-Darcy; Error estimates; Finite element method; Semi-discrete scheme

International Standard Serial Number (ISSN)

1007-5704

Document Type

Article - Journal

Document Version

Citation

File Type

text

Language(s)

English

Rights

© 2026 Elsevier, All rights reserved.

Publication Date

01 Jan 2026

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