arXiv subjects

Thomas Sales

Publications and source records attributed to Thomas Sales.

7 recordsLinked to original sources

arXiv2026

Finite element approximation of the enthalpy formulation for Stefan problems on evolving surfaces

We propose, and analyse, a spatially discrete evolving surface finite element method for the approximation of the enthalpy formulation of the two-phase Stefan problem posed on an evolving surface. Our approach does not rely on mass-lumping and discrete maximum principles. We prove this numerical method is numerically stable, and prove $\mathcal{O}(\sqrt{h})$ error bounds for the temperature in the $L^2_{L^2}$ norm under minimal regularity assumptions by introducing a new projection-type operator. We complement our analysis with discussion on the implementation of this numerical method, where we propose a novel implementation that avoids errors due to numerical quadrature and which has not previously been considered in the literature even in the stationary, flat setting. We also include numerical experiments and experimental order of convergence demonstrations.

math.NA↗

arXiv2026

An iterative approach to a fluid-rigid body interaction problem

We study a novel approach for the existence of solutions to an incompressible fluid-rigid body interaction problem in three dimensions. Our approach introduces an iteration based on a sequence of related problems posed on domains with prescribed evolution. In particular we prove the short-time existence of strong solutions to a system coupling the incompressible Navier--Stokes equations to the ordinary differential equations governing the motion of a rigid body, with no slip boundary conditions on the boundary of the rigid body, provided that the relative density $\frac{\rho}{\rho_B}$, is sufficiently small. We also discuss the use of our iterative approach in numerical methods for the moving boundary problem, and complement this with some numerical experiments in two dimensions which demonstrate the necessity of the smallness assumption on $\frac{\rho}{\rho_B}$.

math.NA↗

arXiv2025

Fully nonlinear second-order mean field games with nondifferentiable Hamiltonians

We analyse fully nonlinear second-order mean field games (MFG) with nondifferentiable Hamiltonians, which take the form of a coupled system of a fully nonlinear Hamilton-Jacobi-Bellman equation and a Kolmogorov-Fokker-Planck partial differential inclusion (PDI) featuring the set-valued subdifferential of the Hamiltonian. We show the existence of solutions of some stationary MFG systems with quite general coupling operators and nonnegative distributional source terms, on general bounded convex domains, under the primary assumptions of uniform ellipticity and the Cordes condition on the diffusion coefficient. The existence proof is founded on an original, and equivalent, reformulation of the PDI as a nonstandard variational inequality (VI), that offers significant flexibility in passages to limits. Furthermore, the uniqueness of the solution of the PDI/VI system is proved in the case of strictly monotone couplings. We then show how the MFG PDI/VI system in the fully nonlinear setting can be obtained as the limit of a sequence of PDE systems with differentiable Hamiltonians, and we give further results on the continuous dependence of the solution.

math.AP↗

arXiv2024

An evolving surface finite element method for the Cahn-Hilliard equation with a logarithmic potential

In this paper we study semi-discrete and fully discrete evolving surface finite element schemes for the Cahn-Hilliard equation with a logarithmic potential. Specifically we consider linear finite elements discretising space and backward Euler time discretisation. Our analysis relies on a specific geometric assumption on the evolution of the surface. Our main results are $L^2_{H^1}$ error bounds for both the semi-discrete and fully discrete schemes, and we provide some numerical results.

math.NA↗

arXiv2024

The evolving surface Cahn-Hilliard equation with a degenerate mobility

We consider the existence of suitable weak solutions to the Cahn-Hilliard equation with a non-constant (degenerate) mobility on a class of evolving surfaces. We also show weak-strong uniqueness for the case of a positive mobility function, and under some further assumptions on the initial data we show uniqueness for a class of strong solutions for a degenerate mobility function.

math.AP↗

arXiv2024

A fully discrete evolving surface finite element method for the Cahn-Hilliard equation with a regular potential

We study two fully discrete evolving surface finite element schemes for the Cahn-Hilliard equation on an evolving surface, given a smooth potential with polynomial growth. In particular we establish optimal order error bounds for a (fully implicit) backward Euler time-discretisation, and an implicit-explicit time-discretisation, with isoparametric surface finite elements discretising space.

math.NA↗

arXiv2024

Navier-Stokes-Cahn-Hilliard equations on evolving surfaces

We derive a system of equations which can be seen as an evolving surface version of the diffuse interface "Model H" of Hohenberg and Halperin (1977). We then consider the well-posedness for the corresponding (tangential) system when one prescribes the evolution of the surface. Well-posedness is proved for smooth potentials in the Cahn-Hilliard equation with polynomial growth, and also for a thermodynamically relevant singular potential.

math.AP↗