Fast Spectral Solvers for PDEs on Triangulated Surfaces
Published:Dec 30, 2025 20:29
•1 min read
•ArXiv
Analysis
This paper addresses the limitations of existing high-order spectral methods for solving PDEs on surfaces, specifically those relying on quadrilateral meshes. It introduces and validates two new high-order strategies for triangulated geometries, extending the applicability of the hierarchical Poincaré-Steklov (HPS) framework. This is significant because it allows for more flexible mesh generation and the ability to handle complex geometries, which is crucial for applications like deforming surfaces and surface evolution problems. The paper's contribution lies in providing efficient and accurate solvers for a broader class of surface geometries.
Key Takeaways
- •Extends the HPS framework to triangulated geometries.
- •Introduces two high-order strategies for triangular elements.
- •Preserves spectral accuracy and efficiency.
- •Applicable to time-dependent and evolving surfaces.
- •Demonstrated through numerical experiments on reaction-diffusion systems and geometry-driven surface evolution.
Reference
“The paper introduces two complementary high-order strategies for triangular elements: a reduced quadrilateralization approach and a triangle based spectral element method based on Dubiner polynomials.”