Geometry-Aware High-Order Approximation for PDEs

Numerical PDE methods based on high-order polynomial interpolation depend critically on where the domain is sampled. I will describe a geometry-aware strategy for constructing stable polynomial nodes directly on arbitrary smooth domains using continuous particle motion and boundary information. On symmetric domains, the resulting configurations reveal a nearby family with exact radial-angular structure, enabling fast methods for approximating optimal node placement. Our methods also admit a broader extension from point values to edge and cell data, leading toward compatible high-order discretizations that preserve fundamental differential identities of the underlying PDE. This is ongoing joint work with Rodrigo Platte, Sébastien Motsch, and Kimball Johnston, all at ASU's SoMSS.