Christian Lubich, Professor, University of Tuebingen, Germany

State Key Laboratory of Scientific and Engineering Computing

Convergent evolving finite element methods for geometric flows

10:00-11:00 September 5(Thursday), N204

Geometric flows of closed surfaces are important in a variety of applications, ranging from the diffusion-driven motion of the surface of a crystal to models for biomembranes and tumor growth. Basic geometric flows are mean curvature flow (described by a spatially second-order evolution equation) and Willmore flow and the closely related surface diffusion flow (described by spatially fourth-order evolution equations). Devising provably convergent surface finite element algorithms for such geometric flows has long remained an open problem, going back to pioneering work by Dziuk in 1988. Recently, a solution to this problem for various geometric flows including those mentioned above was found and explored in joint work with Balázs Kovács and Buyang Li. The proposed algorithms discretize evolution equations for geometric quantities along the flow, in our cases the normal vector and mean curvature, and use these evolving geometric quantities in the velocity law interpolated to the finite element space. This numerical approach admits a convergence analysis, which yields optimal-order H^1-norm error bounds.

Short Bio: Christian Lubich is a professor and head of Numerical Analysis Groups at the University of Tuebingen, Germany. His research concerns with mathematical analysis, numerical analysis, differential equation, exponential integrator and numerical stability. He is an expert on numerical analysis of time-dependent problems, including ordinary/partial differential equations and evolutionary integral equations. His recent research interests include dynamical low-rank approximation of high-dimensional matrix and tensor differential equations with applications to quantum dynamics and plasma physics, highly oscillatory problems, stable numerical interior-exterior coupling of wave equations, and the numerical analysis of geometric evolution equations. He is a famous applied mathematician with many influential works. He won a SIAM Dahlquist Prize, and gave a plenary lecture at the International Congress of Mathematicians in Rio de Janeiro in 2018.