Francis Aznaran

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:
- 17–18th Oct 2025: Finite Element Circus, George Mason University, Virginia.
- 20–24th Jul 2025: US National Congress on Computational Mechanics, Chicago, Illinois, at minisymposium Immersed-boundary variational methods: theory, data structures, and applications.
- 24–27th 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.
- 9th 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:
- 1st prize, Postdoc Lightning Talk Competition, University of Notre Dame College of Science: Numerical modelling of fluid and solid mechanics (19th 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–12th 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.
- Jun 2023: Examination marker, 1st 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, 3rd 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, 3rd year Functional Analysis II.
TA, Masters Finite Element Methods for PDEs. - Michaelmas 2020: TA, 3rd 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, 3rd year Functional Analysis II.
- Michaelmas 2020: Tutor, 1st year Introductory Calculus, Oriel College, Oxford.
Contact:
faznaran@nd.edu203F Crowley Hall, Notre Dame, Indiana, 46556, USA