|Dr. Balazs Kovacs, from University of Tuebingen, gave a lecture titled “A convergent algorithm for mean curvature flow” at AMSS on 13, September 2019.
In the talk, they sketched a proof of convergence was for semi- and full discretizations of mean curvature flow of closed two-dimensional surfaces. The proposed and studied numerical method combined evolving surface finite elements, whose nodes determined the discrete surface like in Dziuk's algorithm proposed in 1990, and linearly implicit backward difference formulae for time integration. The proposed method differed from Dziuk's approach in that it discretized Huisken's evolution equations for the normal vector and mean curvature and uses these evolving geometric quantities in the velocity law projected to the finite element space. This numerical method admitted a convergence analysis, which combined stability estimates and consistency estimates to yield optimal-order -norm error bounds for the computed surface position, velocity, normal vector and mean curvature. The stability analysis was based on the matrix-vector formulation of the finite element method and didn’t not use geometric arguments. The geometry entered only into the consistency estimates. they also presented various numerical experiments to illustrate and complement the theoretical results. Furthermore, they gave an outlook towards problems coupling mean curvature forced by a surface PDE.