Dauphin, Mathias (2025) Polytopal Methods for the Incompressible Variable-Density Navier--Stokes Equations. [Tesi di dottorato]

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Abstract

The present work deals with the numerical approximation of the incompressible variable-density Navier--Stokes (VDNS) equations, a model used for multiphase fluid flows. We employ a new-generation family of numerical methods able to solve differential problems on general polytopal meshes, namely the discontinuous Galerkin (dG) method, the Hybrid High-Order (HHO) method and the Virtual Element Method (VEM). The contributions of this thesis, advancing the application of polytopal methods to multiphase flow simulation, and providing both theoretical foundations and numerical tools, are twofold. In a first part, the whole system of equations is discretized through a hybrid dG-HHO scheme. The semi-discrete scheme is analyzed by means of energy methods and discrete compactness techniques and proved to be convergent towards a weak solution of the continuous problem. The fully-discrete scheme is then obtained through a staggered time-stepping approach, for which well-posedness is proved and convergence rates are confirmed numerically, opening the way to real-world applications. In a second part, we focus on the interface tracking between phases of fluid by developing a stabilized VEM for the steady transport of density. An SUPG-stabilization is introduced to ensure robustness with respect to high velocity fields. A comprehensive analysis establishes well-posedness, derives optimal a priori error estimate and provides an a posteriori analysis for adaptive mesh refinement. Extensive numerical investigations demonstrate the robustness of the scheme and assess its ability to approximate compactly supported density solutions.

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