Distinguished Lecture——BOUNDARY INTEGRAL EXTERIOR CALCULUS
报告人：Ralf Hiptmair (ETH Zürich)
时间：2026-01-10 15:00-16:00
地点：智华楼王选报告厅
报告摘要：
I am going to tell the story of a surprising and deep structural property of boundary integral operators occurring in first-kind boundary integral equations associated with Hodge–Dirac and Hodge-Helmholtz operators for de Rham Hilbert complexes on a bounded domain Ω in a Riemannian manifold. We show that, as regards their induced bilinear forms, those boundary integral operators are Hodge–Dirac and Hodge–Laplace operators in the weak sense, this time set in a trace de Rham Hilbert complex on the boundary∂Ω whose underlying spaces of differential forms are equipped with non-local inner products defined through layer potentials. On the way to this main result we conduct a thorough analysis of layer potentials in operator-induced trace spaces and derive representation formulas.
报告人介绍：
Ralf Hiptmair is a Professor in the Department of Applied Mathematics at ETH Zurich. Since 2008, he has also served as the Director of Studies for the BSc and MSc programs in Computational Science and Engineering (CSE). His primary research interest is the numerical analysis of partial differential equations, with a specific focus on linear partial differential equations, finite element methods (FEM), and boundary element methods (BEM). His work encompasses the design and analysis of mathematical models and numerical methods for computing electromagnetic fields, as well as the development of efficient preconditioners. His recent research also extends to shape calculus and deterministic numerical methods for kinetic equations. Professor Hiptmair received the SIAM Outstanding Paper Prize in 2000 for his work on multigrid methods for Maxwell’s equations. He serves on the editorial boards of several prestigious journals, including the Journal of Computational Mathematics, Zeitschrift für Angewandte Mathematik und Physik (ZAMP), BIT – Numerical Mathematics, Numerische Mathematik, IMA Journal of Numerical Analysis, and Mathematics of Computation.
