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

Gentian Zavalani

Abstract

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.

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

Abstract

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 complexity for repeated solves on meshes with 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.

Paper Structure

This paper contains 13 sections, 1 theorem, 81 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Dubiner polynomials on the reference triangle
  3. High-order parametric surface approximation
  4. High-order spectral collocation on a single element
  5. Domain decomposition methods
  6. Merging Dirichlet-to-Neumann maps
  7. Computational asymptotic complexity
  8. Time-dependent equations
  9. Numerical examples
  10. Laplace–Beltrami problem on the sphere
  11. Surface diffusion equation on the sphere
  12. Numerical study of spatial pattern formation in Turing systems
  13. Appendix. Tangential differential calculus on surfaces

Key Result

Theorem 7.1

\newlabelconv.theorem0 An $s$-step IMEX-BDF scheme, as defined in eq:imex_bdf, cannot achieve an order of accuracy higher than $s$. $\blacktriangleleft$$\blacktriangleleft$

Figures (10)

  • Figure 1: Construction of a surface parametrization over the reference simplex $\Delta_2$ via closest-point projection from the piecewise affine approximation $\Gamma_{h}$.
  • Figure 1: Interface coupling of two surface elements via continuity of the solution and its binormal derivative. Red points indicate boundary collocation points $\mathcal{I}_{b_1}$ and $\mathcal{I}_{b_2}$, while blue points represent interface collocation points $\mathcal{I}_{s_1}$ and $\mathcal{I}_{s_2}$, aligned for coupling. The right panel highlights the shared interface and point alignment used in the spectral collocation framework.
  • Figure 1: A high-order triangular patch mesh is used to discretize the Laplace--Beltrami equation on the upper hemisphere. The spectral discretization exhibits high-order convergence consistent with the rate $\mathcal{O}(h^{n-1})$.
  • Figure 2: A high-order triangulated sphere mesh is used to approximate a spherical harmonic. The observed algebraic decay aligns with the fit rate $h^{n-1}$.
  • Figure 3: (Left) Computed solution at $t = 1$ using time step $\Delta t = 10^{-3}$. (Right) $L_\infty$-error versus polynomial degree for different mesh sizes $h$. The simulations were performed using the implicit--explicit backward differentiation formula (IMEX--BDF4) scheme.
  • ...and 5 more figures

Theorems & Definitions (2)

  • Definition 3.1: Order -$n$ simplex parametrization
  • Theorem 7.1: ascher1995implicit, Theorem 2.1