arXiv subjects

Aaron Brunk

Publications and source records attributed to Aaron Brunk.

At least 19 recordsLinked to original sources

arXiv2026

Generalised dissipative solutions for a non-isothermal phase-field system: existence, weak-strong uniqueness, and long-time behaviour

We study a thermodynamically consistent non-isothermal phase-field system coupling two order parameters and the inverse temperature through a fully non-diagonal Onsager mobility. The model describes the interaction of mass diffusion, heat conduction, and local phase relaxation while conserving mass and internal energy and producing entropy. Global generalised dissipative weak solutions are constructed using a fully discrete approximation. The discrete scheme conserves mass and internal energy and satisfies a discrete entropy inequality. An availability estimate, together with a dimension-adapted barrier, yields strict positivity at fixed mesh and uniform estimates. Compactness then allow passage to the continuous system. Concentration of the singular part of the internal energy is represented by a non-negative defect measure. The entropy-availability structure yields finite dissipation on the infinite time interval and the existence of stationary $\omega$-limit states. Finally, a relative-entropy argument establishes weak-strong uniqueness whenever the weak and strong solutions remain in a bounded thermodynamic state range.

math.AP↗

arXiv2026

High-order conforming finite elements for the Cahn-Hilliard equation: Relative-energy stability and energy defects

We study a semidiscrete single-field Galerkin approximation of the Cahn-Hilliard equation using high-order conforming finite element spaces. More specifically, globally $C^1$ finite elements with $H^2$-conforming trial spaces, including Argyris, Bell, and Bogner-Fox-Schmit elements, allow a direct discretization of the fourth-order formulation and preserve mass exactly. The main structural result is an exact energy balance for the physical Cahn-Hilliard energy. Besides the expected discrete dissipation, the balance contains an explicitly computable energy defect. This defect vanishes for Laplacian-invariant periodic spaces, such as Fourier spaces, but is generally nonzero for classical $C^1$ finite elements. It therefore quantifies the precise loss of a discrete gradient-flow structure. We prove semidiscrete a priori error estimates by a relative-energy argument. The estimate is closed using an augmented relative energy and a discrete elliptic reconstruction bound for the inverse discrete Laplacian. The resulting convergence rates match the expected approximation orders. Numerical experiments with Bell and Argyris elements confirm the rates and demonstrate the defect mechanism by comparison with a Fourier reference discretization.

math.NA↗

arXiv2026

Entropy-stable and energy-conservative fully-discrete finite element method for non-isothermal phase-field models

This work presents a conforming finite-element scheme for non-isothermal phase-field systems coupled to the incompressible Navier-Stokes equations. The proposed numerical scheme preserves entropy production and total energy conservation exactly by variable transformations using entropy as main variable instead of temperature. Convergence tests in space are conducted, and representative examples are provided to demonstrate the scheme's effectiveness.

math.NA↗

arXiv2026

Structure-preserving approximation for non-isothermal phase-field models in melt flow

This work presents a conforming finite-element scheme for the non-isothermal Allen-Cahn-Navier-Stokes system, incorporating periodic, closed, and thermal boundary conditions. The system comprises the incompressible Navier-Stokes equations coupled with the non-isothermal Allen-Cahn equation, which includes a non-conserved phase-field equation and a temperature equation. The proposed numerical scheme preserves entropy production exactly and maintains total energy conservation up to a negative numerical dissipation. Convergence tests in both space and time are conducted, and representative examples are provided to demonstrate the scheme's effectiveness.

math.NA↗

arXiv2026

Phase-field investigation of non-isothermal solidification coupled with melt flow dynamics

Solidification, coupled with melt flow, plays a critical role in determining the microstructure and properties of materials in several manufacturing processes. Phase-field models coupled with the Navier-Stokes equations are widely used to model and simulate these dynamics. However, most existing models neglect essential thermodynamic couplings, particularly the capillary (Korteweg) stress in the momentum equation. This stress, which arises from the coupling between the phase field and the melt flow, accounts for thermal capillary effects during non-isothermal solidification. Neglecting it leads to models inconsistent with non-equilibrium thermodynamics and incapable of capturing capillarity-driven melt flow. In this work, we present a thermodynamically consistent, non-isothermal phase-field model for solidification coupled with melt flow, incorporating cross-coupling terms and explicitly including the Korteweg stress in the momentum equation. Model validation is performed for solidification-only cases, followed by simulations of dendritic growth under melt flow. The results show that thermal capillary effects induce flow near the interface, influencing dendrite tip velocity and morphology. Simulations under forced convection further demonstrate asymmetric dendrite growth due to the imposed flow field. Additionally, we numerically demonstrate the influence of viscosity interpolation schemes on enforcing the no-slip boundary condition in phase-field models with melt flow.

physics.flu-dyn↗

arXiv2026

Review of thermodynamic structures and structure-preserving discretisations of Cahn--Hilliard-type models

The Cahn-Hilliard equation and extensions, notably the Cahn-Hilliard-Darcy and Cahn-Hilliard-Navier-Stokes systems, provide widely used frameworks for coupling interfacial thermodynamics with flow. This review surveys the thermodynamic structures underlying these models, focusing on the formulation of free energy functionals, dissipation mechanisms, and variational principles. We compare structural properties, emphasizing how these models encode conservation laws and energy dissipation. A central theme is the translation of these thermodynamic structures into numerical practice by providing representative discretisation strategies that aim to preserve mass conservation, stability, and energy decay. Particular attention is paid to the trade-offs between accuracy, efficiency, and structure preservation in large-scale simulations.

math.NA↗

arXiv2025

A posteriori existence of strong solutions to the Navier-Stokes equations in 3D

Global existence of strong solutions to the three-dimensional incompressible Navier-Stokes equations remains an open problem. A posteriori existence results offer a way to rigorously verify the existence of strong solutions by ruling out blow-up on a certain time interval, using only numerical solutions. In this work we present such a result for the Navier-Stokes equations subject to periodic boundary conditions, which makes use of a version of the celebrated blow-up criterion in the critical space $L^\infty(L^3)$ by Iskauriaza, Ser\"egin and Shverak (2003). Our approach is based on a conditional stability estimate in $L^2$ and $L^3$. The a posteriori criterion that, if satisfied, verifies existence of strong solutions, involves only negative Sobolev norms of the residual. We apply the criterion to numerical approximations computed with mixed finite elements and an implicit Euler time discretisation. A posteriori error estimates allow us to derive a fully computable criterion without imposing any extra assumptions on the solution. While limited to short time intervals, with sufficient computational resources in principle the criterion might allow for a verification over longer time intervals than what can be achieved by theoretical means.

math.NA↗

arXiv2025

Structure-preserving approximation of the non-isothermal Cahn-Hilliard system based on the entropy equation

We propose and analyze a structure-preserving approximation of the non-isothermal Cahn-Hilliard equation using conforming finite elements for the spatial discretization and a problem-specific mixed explicit-implicit approach for the temporal discretization. To ensure the preservation of structural properties, i.e. conservation of mass and internal energy as well as entropy production, we introduce a suitable variational formulation for the continuous problem, based on the entropy equation. Analytical findings are supported by numerical tests, including convergence analysis.

math.NA↗

arXiv2025

A structure-preserving numerical method for quasi-incompressible Navier-Stokes-Maxwell-Stefan systems

A conforming finite element scheme with mixed explicit-implicit time discretization for quasi-incompressible Navier-Stokes-Maxwell-Stefan systems in a bounded domain with periodic boundary conditions is presented. The system consists of the Navier-Stokes equations, together with a quasi-incompressibility constraint, coupled with the cross-diffusion Maxwell-Stefan equations. The numerical scheme preserves the partial masses and the quasi-incompressibility constraint and dissipates the discrete energy. Numerical experiments in two space dimensions illustrate the convergence of the scheme and the structure-preserving properties.

math.NA↗

arXiv2025

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

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.

math.NA↗

arXiv2025

A simple, fully-discrete, unconditionally energy-stable method for the two-phase Navier-Stokes Cahn-Hilliard model with arbitrary density ratios

The two-phase Navier-Stokes Cahn-Hilliard (NSCH) mixture model is a key framework for simulating multiphase flows with non-matching densities. Developing fully discrete, energy-stable schemes for this model remains challenging, due to the possible presence of negative densities. While various methods have been proposed, ensuring provable energy stability under phase-field modifications, like positive extensions of the density, remains an open problem. We propose a simple, fully discrete, energy-stable method for the NSCH mixture model that ensures stability with respect to the energy functional, where the density in the kinetic energy is positively extended. The method is based on an alternative but equivalent formulation using mass-averaged velocity and volume-fraction-based order parameters, simplifying implementation while preserving theoretical consistency. Numerical results demonstrate that the proposed scheme is robust, accurate, and stable for large density ratios, addressing key challenges in the discretization of NSCH models.

math.NA↗

arXiv2024

Analysis and discretization of the Ohta-Kawasaki equation with forcing and degenerate mobility

The Ohta-Kawasaki equation models the mesoscopic phase separation of immiscible polymer chains that form diblock copolymers, with applications in directed self-assembly for lithography. We perform a mathematical analysis of this model under degenerate mobility and an external force, proving the existence of weak solutions via an approximation scheme for the mobility function. Additionally, we propose a fully discrete scheme for the system and demonstrate the existence and uniqueness of its discrete solution, showing that it inherits essential structural-preserving properties. Finally, we conduct numerical experiments to compare the Ohta-Kawasaki system with the classical Cahn-Hilliard model, highlighting the impact of the repulsion parameter on the phase separation dynamics.

math.NA↗

arXiv2024

Structure-preserving approximation of the Cahn-Hilliard-Biot system

In this work, we propose a structure-preserving discretisation for the recently studied Cahn-Hilliard-Biot system using conforming finite elements in space and problem-adapted explicit-implicit Euler time integration. We prove that the scheme preserves the thermodynamic structure, that is, the balance of mass and volumetric fluid content and the energy dissipation balance. The existence of discrete solutions is established under suitable growth conditions. Furthermore, it is shown that the algorithm can be realised as a splitting method, that is, decoupling the Cahn-Hilliard subsystem from the poro-elasticity subsystem, while the first one is nonlinear and the second subsystem is linear. The schemes are illustrated by numerical examples and a convergence test.

math.NA↗

arXiv2024

Error analysis for a viscoelastic phase separation model

We consider systematic numerical approximation of a viscoelastic phase separation model that describes the demixing of a polymer solvent mixture. An unconditionally stable discretisation method is proposed based on a finite element approximation in space and a variational time discretization strategy. The proposed method preserves the energy-dissipation structure of the underlying system exactly and allows to establish a fully discrete nonlinear stability estimate in natural norms based on the concept of relative energy. These estimates are used to derive order optimal error estimates for the method under minimal smoothness assumptions on the problem data, despite the presence of various strong nonlinearities in the equations. The theoretical results and main properties of the method are illustrated by numerical simulations which also demonstrate the capability to reproduce the relevant physical effects observed in experiments.

math.NA↗

arXiv2024

Nonisothermal Cahn-Hilliard Navier-Stokes system

In this research, we introduce and investigate an approximation method that preserves the structural integrity of the non-isothermal Cahn-Hilliard-Navier-Stokes system. Our approach extends a previously proposed technique [1], which utilizes conforming (inf-sup stable) finite elements in space, coupled with implicit time discretization employing convex-concave splitting. Expanding upon this method, we incorporate the unstable P1|P1 pair for the Navier-Stokes contributions, integrating Brezzi-Pitk\"aranta stabilization. Additionally, we improve the enforcement of incompressibility conditions through grad div stabilization. While these techniques are well-established for Navier-Stokes equations, it becomes apparent that for non-isothermal models, they introduce additional coupling terms to the equation governing internal energy. To ensure the conservation of total energy and maintain entropy production, these stabilization terms are appropriately integrated into the internal energy equation.

math.NA↗

arXiv2024

Robust a posteriori error control for the Allen-Cahn equation with variable mobility

In this work, we derive a $\gamma$-robust a posteriori error estimator for finite element approximations of the Allen-Cahn equation with variable non-degenerate mobility. The estimator utilizes spectral estimates for the linearized steady part of the differential operator as well as a conditional stability estimate based on a weighted sum of Bregman distances, based on the energy and a functional related to the mobility. A suitable reconstruction of the numerical solution in the stability estimate leads to a fully computable estimator.

math.NA↗

arXiv2024

Structure-preserving approximation for the non-isothermal Cahn-Hilliard-Navier-Stokes system

In this work we propose and analyse a structure-preserving approximation of the non-isothermal Cahn-Hilliard-Navier-Stokes system using conforming finite elements in space and implicit time discretisation with convex-concave splitting. The system is first reformulated into a variational form which reveal the structure of the equations, which is then used in the subsequent approximation.

math.NA↗

arXiv2023

Variational approximation for a non-isothermal coupled phase-field system: Structure-preservation & Nonlinear stability

A Cahn-Hilliard-Allen-Cahn phase-field model coupled with a heat transfer equation, particularly with full non-diagonal mobility matrices, is studied. After reformulating the problem w.r.t. the inverse of temperature, we proposed and analysed a structure-preserving approximation for the semi-discretisation in space and then a fully discrete approximation using conforming finite elements and time-stepping methods. We prove structure-preserving property and discrete stability using relative entropy methods for the semi-discrete and fully discrete case. The theoretical results are illustrated by numerical experiments.

math.NA↗