A High-Order Fast Direct Solver for Surface PDEs on Triangles

TL;DR

This paper introduces a triangle-based hierarchical Poincaré-Steklov method for efficiently solving elliptic PDEs on surfaces with unstructured meshes, achieving high-order accuracy and $O(N \,\log N)$ complexity.

Contribution

It extends the classical HPS method to unstructured triangular meshes using orthogonal Dubiner bases, enabling high-order discretizations on complex geometries.

Findings

Achieves spectral accuracy and high-order convergence.

Provides a fast direct solver with $O(N \,\log N)$ complexity.

Effective for implicit time-stepping of surface PDEs.

Abstract

We develop a triangular formulation of the hierarchical Poincar\'e-Steklov (HPS) method for elliptic partial differential equations on surfaces, allowing high-order discretizations on unstructured meshes and complex geometries. Classical HPS formulations rely on high-order quadrilateral meshes and tensor-product spectral discretizations, which enable efficient algorithms but restrict applicability to structured geometries. To overcome this restriction, we introduce a triangle-based hierarchical Poincar\'e-Steklov scheme (THPS) built on orthogonal Dubiner polynomial bases. As in the classical HPS framework, local solution operators and Dirichlet-to-Neumann maps are constructed and merged hierarchically, yielding a fast direct solver with $O(N \log N)$ complexity for repeated solves on meshes with $N$ elements. The reuse of precomputed operators makes the method particularly effective for implicit time-stepping of surface PDEs. Numerical experiments demonstrate that the proposed method retains spectral accuracy and achieves high-order convergence for a range of static and time-dependent test problems.