A reduced order model for a stable embedded boundary parametrized Cahn-Hilliard phase-field system based on cut finite elements

TL;DR

This paper develops a reduced order cut finite element method for the parametrized Cahn-Hilliard system, effectively handling nonlinearities and geometric variations without remeshing, validated through numerical experiments.

Contribution

It introduces a novel reduced order model based on POD-Galerkin for a stabilized cut finite element approach to the parametrized Cahn-Hilliard system, improving efficiency and robustness.

Findings

Effective stabilization of nonlinearities in cut finite element methods.

Robust reduced order basis independent of geometric changes.

Numerical validation with pseudo-random initial data.

Abstract

In the present work, we investigate a cut finite element method for the parameterized system of second-order equations stemming from the splitting approach of a fourth order nonlinear geometrical PDE, namely the Cahn-Hilliard system. We manage to tackle the instability issues of such methods whenever strong nonlinearities appear and to utilize their flexibility of the fixed background geometry -- and mesh -- characteristic, through which, one can avoid e.g. in parametrized geometries the remeshing on the full order level, as well as, transformations to reference geometries on the reduced level. As a final goal, we manage to find an efficient global, concerning the geometrical manifold, and independent of geometrical changes, reduced order basis. The POD-Galerkin approach exhibits its strength even with pseudo-random discontinuous initial data verified by numerical experiments.