Francis Aznaran

Society of Science Research Fellow in Applied and Computational Mathematics,
University of Notre Dame.

STUDENTS OF ACMS/MATH 40390 NUMERICAL ANALYSIS, SPRING 2025: All course information, and homework and computer project assignment and submission, is via the course Canvas page.

Research interests:

The finite element method for the PDEs of elasticity, fluids, and thermodynamics, encompassing:

• Piola transformation theory and implementation of finite element bases
• Elasticity complexes and the discretisation of elastic stress and strain
• Multicomponent fluid flows
• Fluid-structure interaction

Preprints:

• Aznaran, F. R. A., Hu, K., and Parker II, C., Uniformly hp-stable elements for the elasticity complex, arXiv:2409.17414 (2024) [pdf, bib]
• Aznaran, F. R. A., Bukač, M., Muha, B., and Salgado, A. J., Analysis and finite element approximation of a diffuse interface approach to the Stokes–Biot coupling, arXiv:2407.05949 (2024) [pdf, bib]

Publications:

• Aznaran, F. R. A., Farrell, P. E., Monroe, C. W., and Van-Brunt, A. J., Finite element methods for multicomponent convection-diffusion, IMA Journal of Numerical Analysis (2024), pp. 1–35, DOI: 10.1093/imanum/drae001 [pdf, bib]
• Aznaran, F. R. A., Farrell, P. E., Kirby, R. C., Transformations for Piola-mapped elements, SMAI Journal of Computational Mathematics, Vol. 8 (2022), pp. 399–437, DOI: 10.5802/smai-jcm.91 [pdf, bib]

Forthcoming talks:

• 17–18^th Oct 2025: Finite Element Circus, George Mason University, Virginia.
• 20–24^th Jul 2025: US National Congress on Computational Mechanics, Chicago, Illinois, at minisymposium Immersed-boundary variational methods: theory, data structures, and applications.
• 24–27^th Jun 2025: Organiser and speaker, minisymposium The finite element method for problems in multiphysics and geometry, at the Biennial Numerical Analysis Conference, University of Strathclyde, Scotland.
• 9^th Jun 2025: International Conference on Adaptive Modelling and Simulation, Barcelona, at minisymposium Adaptive modeling and simulation by immersed boundary methods.

Brief CV:

• 2023–present: Society of Science Research Fellow in Applied and Computational Mathematics and Statistics, University of Notre Dame, Indiana, USA.
• 2022–23: EPSRC Postdoctoral Research Associate (independent), Mathematical Institute, University of Oxford, UK.
• Mar–Apr 2023: Visiting Scholar, Department of Mathematics, Baylor University, Texas, USA.
• May 2023: Visitor, Technische Universität Kaiserslautern, Germany.
• 2018–22: DPhil candidate in Numerical Analysis/OxPDE, University of Oxford, UK (via the EPSRC Centre for Doctoral Training in PDEs). The Queen's College.
• Thesis: Discretisation of Hodge Laplacians in the elasticity complex.
• Jun–Aug 2021: Software engineering intern, The MathWorks, Cambridge, UK.
• Jul–Aug 2020: Software engineering intern, The MathWorks, Cambridge, UK.
• 2017–18: MASt Applied Mathematics (Part III), University of Cambridge, UK. Corpus Christi College.
• Essay: Highly oscillatory quadrature.
• 2014–17: BSc Mathematics, University of Warwick, UK.

Awards:

• 1^st prize, Postdoc Lightning Talk Competition, University of Notre Dame College of Science: Numerical modelling of fluid and solid mechanics (19^th Sept 2024).
• Combined EPSRC/MathWorks Scholarship at the PDEs CDT, Oxford Mathematical Institute (2018–22).

Software:

User of the Firedrake finite element software, in which implemented the conforming and nonconforming Arnold–Winther, Mardal–Tai–Winther, and Hu–Zhang finite elements.

Further information:

• Math circles for pre-K–12^th grade students, via Riverbend Community Math Center, South Bend, IN; with Prof. Brian Hall.
• Notre Dame College of Science Postdoc Spotlight.
• Oxford FEM reading group is now organised by Charles Parker II; see here for texts covered Jan 2020 – Aug 2023.

Teaching:

At University of Notre Dame:

• Spring 2025: Instructor of Record, senior 3-credit ACMS/MATH 40390 Numerical Analysis.

At University of Oxford:

• Jun 2023: Examination marker, 1^st year Constructive Mathematics.
  Assessor (double blind), Masters Case Studies in Scientific Computing.
  Assessor (double blind), Masters dissertations in analysis.
• Hilary 2023: Tutor, Masters Finite Element Methods for PDEs.
• Dec 2022: Interviewer, undergraduate mathematics admissions, St Edmund Hall, Oxford.
• Michaelmas 2022: Tutor, 3^rd year Functional Analysis I.
• Hilary 2022: Tutor, Masters Fixed Point Methods for Nonlinear PDEs.
• Dec 2021: Interviewer, undergraduate mathematics admissions, Worcester College, Oxford.
• Michaelmas 2021: Tutor, Masters Numerical Solution of PDEs.
• Hilary 2021: Tutor, 3^rd year Functional Analysis II.
  TA, Masters Finite Element Methods for PDEs.
• Michaelmas 2020: TA, 3^rd year Functional Analytic Methods for PDEs.
  TA, Masters Numerical Solution of Differential Equations I.
• Nov 2020: Examination marker, Oxford Maths Admissions Test.
• Hilary 2020: TA, 3^rd year Functional Analysis II.
• Michaelmas 2020: Tutor, 1^st year Introductory Calculus, Oriel College, Oxford.

Contact:

faznaran@nd.edu
203F Crowley Hall, Notre Dame, Indiana, 46556, USA