Work

Analysis and structure-preserving approximation of a Cahn–Hilliard–Forchheimer system with solution-dependent mass and volume source

Year: 2025

Type: article

Abstract: We analyze a coupled Cahn–Hilliard–Forchheimer system featuring concentration-dependent mobility, mass source and convective transport. The velocity field is governed by a generalized quasi-incompressible Forchheimer equation with solution-dependent volume source. We impose Dirichlet boundary conditions for the pressure to accommodate the source term. Our contributions include a novel well-posedness result for the generalized Forchheimer subsystem via the Browder–Minty theorem, and existence of weak solutions for the full coupled system established through energy estimates at the Galerkin level combined with compactness techniques such as Aubin–Lions’ lemma and Minty’s trick. Furthermore, we develop a structure-preserving discretization using Raviart–Thomas elements for the velocity that maintains exact mass balance and discrete energy-dissipation balance, with well-posedness demonstrated through relative energy estimates and inf-sup stability. Lastly, we validate our model through numerical experiments, demonstrating optimal convergence rates, structure preservation, and the role of the Forchheimer nonlinearity in governing phase-field evolution dynamics.

Source: ESAIM Mathematical Modelling and Numerical Analysis

DOI: 10.1051/m2an/2025081

Authors: Aaron Brunk, Marvin Fritz

Institutions: Johannes Gutenberg University Mainz, Johann Radon Institute for Computational and Applied Mathematics

Language: English

Topic: Advanced Numerical Methods in Computational Mathematics

Subfield: Computational Mechanics

Field: Engineering

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid:

Funders: Deutsche Forschungsgemeinschaft, Gutenberg Forschungskolleg