Aaron Brunk

Analysis of PDEs, Numerical Analysis, Mathematical Modeling.

Most of my preprints can be found on arXiv.org and ResearchGate. My articles can also be tracked on Google Scholar and ORCID.

Preprints:

  1. Structure-preserving approximation of the non-isothermal Cahn-Hilliard system
    with Mária Lukácová-Medvidová, Dennis Schumann
    Links: Preprint
  2. A structure-preserving numerical method for quasi-incompressible Navier–Stokes–Maxwell–Stefan systems
    with Ansgar Jüngel, Mária Lukácová-Medvidová
    Links: Preprint
  3. Analysis and structure-preserving approximation of a Cahn-Hilliard-Forchheimer system with solution-dependent mass and volume source
    with Marvin Fritz
    Links: Preprint
  4. A simple, fully-discrete, unconditionally energy-stable method for the two-phase Navier-Stokes Cahn-Hilliard model with arbitrary density ratios
    with Marco F. P. ten Eikelder
    Links: Preprint
  5. Analysis and discretization of the Ohta-Kawasaki equation with forcing and degenerate mobility
    with Marvin Fritz
    Links: Preprint
  6. Fundamentals of the Oldroyd-B model revisited: Tensorial vs. vectorial theory
    with Joydip Chaudhuri, Burkhard Dünweg and Mária Lukácová-Medvidová
    Links: Preprint

Peer-Reviewed Journal Articles:

  1. Error analysis for a second order approximation of a viscoelastic phase separation model
    with Herbert Egger, Oliver Habrich, Mária Lukácová-Medvidová
    Numerische Mathematik (2025)
    Links: Preprint & Journal

  2. Robust a posteriori error control for the Allen-Cahn equation with variable mobility
    with Jan Giesselmann and Mária Lukácová-Medvidová
    SIAM Journal on Numerical Analysis (2025)
    Links: Preprint & Journal

  3. Structure-preserving approximation of the Cahn-Hilliard-Biot system
    with Marvin Fritz
    Numerical Methods for Partial Differential Equations (2024)
    Links: Preprint & Journal

  4. A second-order structure-preserving discretization for the Cahn-Hilliard/Allen-Cahn system with cross-kinetic coupling
    with Herbert Egger, Oliver Habrich
    Applied Numerical Mathematics (2024)
    Links: Preprint & Journal

  5. Variational approximation for a non-isothermal coupled phase-field system: Structure-preservation & Nonlinear stability
    with Oliver Habrich, Timileyin David Oyedeji, Yangyiwei Yang, Bai-Xiang Xu
    Computational Methods in Applied Mathematics (2024)
    Links: Preprint & Journal

  6. On existence, uniqueness and stability of solutions to Cahn-Hilliard/Allen-Cahn systems with cross-kinetic coupling
    with Herbert Egger, Timileyin David Oyedeji, Yangyiwei Yang and Bai-Xiang Xu
    Nonlinear Analysis: Real World Applications (2023)
    Links: Preprint & Journal

  7. A second-order fully-balanced structure-preserving variational discretization scheme for the Cahn-Hilliard Navier-Stokes system
    with Herbert Egger, Oliver Habrich and Mária Lukácová-Medvidová
    Mathematical Models and Methods in Applied Sciences (M3AS) (2023)
    Links: Preprint & Journal

  8. On uniqueness and stable estimation of multiple parameters in the Cahn-Hilliard equation
    with Herbert Egger, Oliver Habrich
    Inverse Probleme (2023)
    Links: Preprint & Journal

  9. Stability and discretization error analysis for the Cahn-Hilliard system via relative energy estimates
    with Herbert Egger, Oliver Habrich and Mária Lukácová-Medvidová
    ESAIM: Mathematical Modelling and Numerical Analysis (2023)
    Links: Preprint & Journal

  10. Existence and weak-strong uniqueness for global weak solutions for the viscoelastic phase separation model in three space dimensions
    Discrete and Continuous Dynamical Systems (2023)
    Links: Preprint & Journal

  11. Relative energy and weak-strong uniqueness of the two-phase viscoelastic phase separation model
    with Mária Lukácová-Medvidová
    Zeitschrift für Angewandte Mathematik und Mechanik (2022)
    Links: Preprint & Journal

  12. Global existence of weak solutions to viscoelastic phase separation: Part II Degenerate Case
    with Mária Lukácová-Medvidová
    Nonlinearity (2022)
    Links: Preprint & Journal

  13. Global existence of weak solutions to viscoelastic phase separation: Part I Regular Case
    with Mária Lukácová-Medvidová
    Nonlinearity (2022)
    Links: Preprint & Journal

  14. Existence, regularity and weak-strong uniqueness for three-dimensional Peterlin viscoelastic model
    with Yong Lu and Mária Lukácová-Medvidová
    Communications in Mathematical Sciences (2022)
    Links: Preprint & Journal

  15. Systematic derivation of hydrodynamic equations for viscoelastic phase separation
    with Burkhard Dünweg, Herbert Egger, Oliver Habrich, Mária Lukácová-Medvidová and Dominic Spiller
    Journal of Physics: Condensed Matter (2021)
    Links: Preprint & Journal

  16. Analysis of a viscoelastic phase separation model
    with Burkhard Dünweg, Herbert Egger, Oliver Habrich, Mária Lukácová-Medvidová and Dominic Spiller
    Journal of Physics: Condensed Matter (2021)
    Links: Preprint & Journal

  17. Modelling cell-cell collision and adhesion with the filament based lamellipodium model
    with Diane Peurichard, Nikolas Sfakianakis and Christian Schmeiser
    Biomath (2018)
    Links: Preprint & Journal

  18. Stability, Convergence, and Sensitivity Analysis of the FBLM and the Corresponding FEM
    with Nikolas Sfakianakis
    Bulletin of Mathematical Biology (2018)
    Links: Preprint & Journal

Peer-Reviewed Proceedings Articles:

  1. Structure-preserving approximation for the non-isothermal Cahn-Hilliard-Navier-Stokes system
    with Dennis Schumann
    Numerical Mathematics and Advanced Applications ENUMATH 2023 (2025)
    Links: Preprint & Journal

  2. Nonisothermal Cahn-Hilliard Navier-Stokes system
    with Dennis Schumann
    Proceedings in Applied Mathematics and Mechanics (2024)
    Links: Preprint & Journal

  3. Chemotaxis and Haptotaxis on Cellular Level
    with Niklas Kolbe and Nikolas Sfakianakis
    HYP 16: Theory, Numerics and Applications of Hyperbolic Problem I (2018)
    Links: Preprint & Journal

Theses:

  • Viscoelastic phase separation: Well-posedness and numerical analysis
    PhD Dissertation, Johannes Gutenberg University Mainz (2022)
  • Numerische Behandlung von zeitgebrochenen Diffusionsgleichungen
    Master’s Thesis, Johannes Gutenberg University Mainz (2017)
  • Mathematische Modellierung von Phosphorylierungssystemen
    Bachelor’s Thesis, Johannes Gutenberg University Mainz (2015)

Current projects/drafts (only on request):

  1. A posteriori existence of strong solutions for the Navier-Stokes equation in 3D with Jan Giesselmann and Tabea Tscherpel, TU Darmstadt
  2. Existence of discrete solutions and optimal errors estimates for quasi-incompressible Navier–Stokes–Maxwell–Stefan systems with Ansgar Jüngel and Mária Lukácová-Medvidová, Tu Vienna/JGU Mainz
  3. Structure-preserving approximations for viscoelastic systems with phase-field dependent energy with Dennis Trautwein, University Regensburg
  4. Structure-preserving approximation for non-isothermal phase-field models in melt flow with Dennis Schumann, JGU Mainz
  5. Modeling and structure-preserving discretization of N-phase incompressible fluid mixtures with arbitrary density ratios with Marco ten Eikelder, TU Darmstadt
  6. Modeling and stable discretisations for Maxwell–Stefan with quasi-incompressible, heat conducting fluids with Patrick Farrell and Mária Lukácová-Medvidová, Oxford/JGU Mainz
  7. Existence and weak-strong uniqueness for non-isothermal phase-field systems with Marvin Fritz, RICAM
  8. Energy-stable H^2 conforming discretisations for the Cahn-Hilliard equation with variable mobility with Marvin Fritz, RICAM

To-Do list (if you are interested just let me know):

  1. Relative energy estimates for non-isothermal phase-field systems including fluid flow
  2. Error estimates for an energy-stable FE discretisation for the Abels-Garcke-Grün model
  3. Comparison between the Abels-Garcke-Grün model and the quasi-incompressible Navier-Stokes-Cahn-Hilliard system
  4. Error estimate for a stable FV scheme for compressible Navier-Stokes
  5. Error estimate for stable FV scheme for cross-diffusion systems
  6. Stable approximations for non-isothermal viscoelastic rate-type fluids with stress diffusion