A High-Order Fast Direct Solver for Surface PDEs on Triangles
This work addresses a domain-specific problem for computational scientists and engineers dealing with surface PDEs on unstructured meshes, offering an incremental improvement by extending an existing method to more flexible geometries.
The paper tackled the problem of solving elliptic partial differential equations on surfaces with complex geometries by developing a triangular formulation of the hierarchical Poincaré-Steklov method, enabling high-order discretizations on unstructured meshes and achieving spectral accuracy with O(N log N) complexity for repeated solves.
We develop a triangular formulation of the hierarchical Poincaré-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 Poincaré-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.