Hello!

Numerical Analysis · Surface PDEs · Fast Direct Solvers

My name is Gentian Zavalani. I am currently a Wrap-up Postdoc at the Institute of Numerical Mathematics, TU Dresden, Germany. I completed my PhD in March 2026 under the supervision of Oliver Sander and Michael Hecht.

I develop and analyze numerical methods for partial differential equations (PDEs) posed on smooth two-dimensional surfaces. My work is motivated by applications in biology and physics and aims at designing fast and accurate algorithms for complex geometries.

Research themes

High-order approximation & quadrature

Super-algebraic integration and interpolation on curved, triangulated surfaces.

Fast direct solvers

Hierarchical Poincaré–Steklov (HPS) solvers for elliptic PDEs on general surfaces.

Dynamics on evolving surfaces

Computational methods for pattern formation and flows on moving geometries.

Baseline Turing pattern — **Interacting Turing systems.** Left to right: baseline pattern, and patterns produced by linear, quadratic, and cubic coupling.

Pattern with linear coupling — **Interacting Turing systems.** Left to right: baseline pattern, and patterns produced by linear, quadratic, and cubic coupling.

Surface vortex roll-up with HPS.
High-order HPS simulation of incompressible vorticity dynamics on a spherical surface. A perturbed equatorial vortex sheet self-organizes into coherent moving vortices.

Teaching

📘 Winter Semester 2025/26: I taught a course on Approximation Theory at TU Dresden, covering the theoretical foundations and practical aspects of function approximation and interpolation — high-order polynomials, trigonometric series, and rational functions.

📝 Lecture notes: “Notes on Approximation Theory”