Fast high-order spectral solvers for PDEs on triangulated surfaces with applications to deforming surfaces

Abstract

In this paper, we extend the classical quadrilateral based hierarchical Poincar\'e-Steklov (HPS) framework to triangulated geometries. Traditionally, the HPS method takes as input an unstructured, high-order quadrilateral mesh and relies on tensor-product spectral discretizations on each element. To overcome this restriction, we introduce two complementary high-order strategies for triangular elements: a reduced quadrilateralization approach which is straightforward to implement, and triangle based spectral element method based on Dubiner polynomials. We show numerically that these extensions preserve the spectral accuracy, efficiency, and fast direct-solver structure of the HPS framework. The method is further extended to time dependent and evolving surfaces, and its performance is demonstrated through numerical experiments on reaction-diffusion systems, and geometry driven surface evolution.