Academic Career

Research Interests

• Eigenvalues and eigenfunctions of differential operators
• Isogeometric Analysis on domains with singularities, local mesh refinement
• Spectral theory and application to the isospectral problem
• Machine Learning for solving inverse problems

Honors and Awards

2023	UKIE SIAM Student Prize, best student presentations at the 29th Biennial Numerical Analysis Conference

Publications

Articles in Peer-Reviewed International Journals

1. Apel, T., Zilk, P. (2024): Isogeometric analysis of the Laplace eigenvalue problem on circular sectors: Regularity properties and graded meshes, Computers and Mathematics With Applications, 175:236–254, DOI (Open Access) arXiv 

Preprints and Articles Submitted for Publication

1. Apel, T., Zilk, P. (2025). Error Estimates and Graded Mesh Refinement for Isogeometric Analysis on Polar Domains with Corners. Preprint, submitted for publication arXiv 
2. Salcedo Lagunero, R., Fellner, K., Apel, T., Kempf, V., Zilk, P. (2024): Lipolysis on Lipid Droplets: Mathematical Modelling and Numerical Discretisation, Preprint, submitted for publication, arXiv 

International Conference and Seminar Contributions with Abstract

1. Zilk, P.: Can one hear the shape of a crack in a drum? – An analytical and data-based approach, 95th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM), Posen, Poland, April 7 - 11, 2025
2. Zilk, P.: IGA on polar domains with corners: graded meshes, collocation points and application to crack identification, invited, Computational Mechanics and Advanced Materials Group - Seminar, Pavia, Italy, February 12, 2025
3. Zilk, P.: Isogeometric analysis on polar domains with corners: Graded meshes and error estimates, Joint GNCS-SIAM Chapters Meeting for Young Researchers in Numerical Analysis and Applied Mathematics, Pavia, Italy, February 10-11, 2025
4. Zilk, P.: Error estimates for isogeometric analysis on polar domains with corners, GAMM Workshop on Numerical Analysis, Augsburg, Germany, November 13 - 15, 2024
5. Zilk, P.: Local isogeometric mesh refinement in polar-type domains with corners, 37th Chemnitz Finite Element Symposium, Chemnitz, Germany, September 9 - 11, 2024
6. Zilk, P.: Graded isogeometric mesh refinement towards corners of polar-type domains, selected, Lake Como Summer School on Isogeometric Analysis: Theory, Applications, and New Trends, Como, Italy, July 8 - 12, 2024
7. Zilk, P.: Can one hear the shape of a crack? Gaining insights through spectral geometry, simulations and machine learning, invited, Numerics research seminar, Chemnitz, Germany, June 26, 2024
8. Zilk, P.: Approximation of Laplace eigenvalues and eigenfunctions of domains with corners, 10th International Conference on Computational Methods in Applied Mathematics (CMAM-10), Bonn, Germany, June 10-14, 2024
9. Zilk, P.: Graded isogeometric mesh refinement towards corners of singularly parametrized domains, 9th European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS Congress 2024), Lisbon, Portugal, June 3 - 7, 2024
10. Zilk, P.: Identifying cracks in membranes via their eigenfrequencies - An analytical and data-based approach, GAMM CSE Workshop, November 10/17, 2023 (virtual)
11. Zilk, P.: Identifying cracks in membranes via their eigenfrequencies - A theoretical and practical approach, 36th Chemnitz Finite Element Symposium, Leibnitz, Austria, September 11 - 13, 2023
12. Zilk, P.: Approximating Laplace eigenvalues of circular sectors using isogeometric mesh grading, 29th Biennial Numerical Analysis Conference, Glasgow, Scotland, June 27-30, 2023
13. Zilk, P.: Isogeometric mesh grading for Laplace eigenvalue problems on circular sectors, 11th International Conference on Isogeometric Analysis (IGA 2023), Lyon, France, June 18-21, 2023
14. Zilk, P.: Introduction to Isogeometric Analysis and its efficiency for solving Laplace eigenvalue problems, invited, CERAMATHS - DMATHS seminar, Valenciennes, France, March 16, 2023
15. Zilk, P.: Isogeometric approximation of eigenfrequencies of membranes with cracks and application to shape identification, 35th Chemnitz Finite Element Symposium, Herrsching am Ammersee, Germany, September 15 - 17, 2022
16. Zilk, P.,  Eckardt, L., Haubner, C., Apel,T. (2021): Numerical study of PDE eigenvalue problems, ECCOMAS Virtual International Conference on Isogeometric Analysis (VIGA), Lyon/Online Conference, September 26-29, 2021

Posters

1. Zilk, P.,  Eckardt, L., Haubner, C., Apel, T. (2021): Numerical study of PDE eigenvalue problems, ECCOMAS Virtual International Conference on Isogeometric Analysis (VIGA), Lyon/Online Conference, September 26-29, 2021, PDF

Monographs

1. Zilk, P. (2021): Flächen-Cluster und offene Flächen mit minimaler Willmore-Energie (Surface-cluster and open surfaces with minimal Willmore-Energy), Master's thesis, University of Regensburg
2. Zilk, P. (2018): Schwache Lösungstheorie für das Kirchhoff-Love-Modell für dünne Platten (Weak solution theory for the Kirchhoff-Love-Model for thin plates), Bachelor's thesis, University of Regensburg

Teaching

Courses & Exercises

• Exercises Differential Equations (FT 2021, FT 2022)
• Exercises Partial Differential Equations (WT 2022, WT 2023)
• Exercises Mathematics I (HT 2022, HT 2023, HT 2024)
• Exercises Mathematics II (HT 2022, HT 2023, HT 2024)
• Exercises Mathematics III (WT 2024, WT 2025)

Supervised Student Projects / Theses

• Locally refined isogeometric collocation methods, Bachelor's thesis (2024)
• Solving eigenvalue problems with isogeometric collocation methods, Student project (2024)
• Isogeometric simulation of eigenfrequencies of domains with cracks using locally refined meshes and application for crack identification, Master's thesis (2023)
• Comparison of models for approximating eigenfrequencies of thin plates, Student project (2023)
• Adaptive interpolation using neural networks, Bachelor's thesis (2021)