Seminar| Institute of Mathematical Sciences
Time: Friday, September 25th, 2026,10:30-11:30
Location: IMS RS408
Speaker: Qing Xia, Wenzhou-Kean University
Abstract: We develop a structure-preserving, ghost-penalty-free CutFEM discretization of planar Steklov eigenvalue problems for both bounded domains and exterior obstacles in complex geometry. Unfitted elements on a Cartesian mesh are paired with a matched chord reconstruction of the interface, yielding a single polygonal discrete geometry. The bulk is eliminated by a Schur reduction to a discrete Dirichlet-to-Neumann operator on the cut band, then restricted to a discrete-harmonic trial space generated by the lattice Green’s function (LGF). The reduced boundary eigenproblem is obtained by a symmetric congruence of the unstabilized CutFEM forms: the energy operator is symmetric and positive semi-definite with constant kernel, while with trace coordinates or a multi-component Calderón right-preconditioner the boundary mass is symmetric positive definite, so the Steklov eigenvalues are real and nonnegative. An LGF bulk extension removes the volumetric stiffness inverse; the same LGF supplies an exact-lattice outer Dirichlet-to-Dirichlet map for exterior problems on a truncated box, without PML or approximate radiation conditions. Numerical tests on disks, annuli, ellipses, nonconvex flowers, and multi-obstacle exteriors show second-order eigenvalue accuracy and controlled conditioning under refinement for the recommended formulations.