A High-Order Fast Direct Solver for Surface PDEs

IF 3 2区数学 Q1 MATHEMATICS, APPLIED

SIAM Journal on Scientific Computing Pub Date : 2024-08-13 DOI:10.1137/22m1525259

Daniel Fortunato

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引用次数: 0

Abstract

SIAM Journal on Scientific Computing, Volume 46, Issue 4, Page A2582-A2606, August 2024.
Abstract. We introduce a fast direct solver for variable-coefficient elliptic PDEs on surfaces based on the hierarchical Poincaré–Steklov method. The method takes as input an unstructured, high-order quadrilateral mesh of a surface and discretizes surface differential operators on each element using a high-order spectral collocation scheme. Elemental solution operators and Dirichlet-to-Neumann maps tangent to the surface are precomputed and merged in a pairwise fashion to yield a hierarchy of solution operators that may be applied in [math] operations for a mesh with [math] degrees of freedom. The resulting fast direct solver may be used to accelerate high-order implicit time-stepping schemes, as the precomputed operators can be reused for fast elliptic solves on surfaces. On a standard laptop, precomputation for a 12th-order surface mesh with over 1 million degrees of freedom takes 10 seconds, while subsequent solves take only 0.25 seconds. We apply the method to a range of problems on both smooth surfaces and surfaces with sharp corners and edges, including the static Laplace–Beltrami problem, the Hodge decomposition of a tangential vector field, and some time-dependent nonlinear reaction-diffusion systems. Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: code and data available”, as a recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at https://github.com/danfortunato/surface-hps-sisc.

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曲面 PDE 的高阶快速直接求解器

SIAM 科学计算期刊》，第 46 卷第 4 期，第 A2582-A2606 页，2024 年 8 月。摘要。我们介绍了一种基于分层 Poincaré-Steklov 方法的曲面上变系数椭圆 PDE 快速直接求解器。该方法以表面的非结构化高阶四边形网格为输入，使用高阶谱配位方案对每个元素上的表面微分算子进行离散化。预先计算元素求解算子和与曲面相切的 Dirichlet 到 Neumann 映射，并以成对的方式进行合并，以生成求解算子的层次结构，可用于具有 [math] 自由度的网格的 [math] 操作。由此产生的快速直接求解器可用于加速高阶隐式时间步进方案，因为预计算算子可重复用于曲面上的快速椭圆求解。在标准笔记本电脑上，对超过 100 万自由度的 12 阶曲面网格进行预计算需要 10 秒，而后续求解只需 0.25 秒。我们将该方法应用于一系列光滑表面和尖角及边缘表面的问题，包括静态拉普拉斯-贝尔特拉米问题、切向矢量场的霍奇分解，以及一些随时间变化的非线性反应扩散系统。计算结果的可重复性。本文被授予 "SIAM 可重复性徽章：代码和数据可用"，以表彰作者遵循了 SISC 和科学计算界重视的可重复性原则。读者可通过 https://github.com/danfortunato/surface-hps-sisc 获取代码和数据，以重现本文中的结果。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

SIAM Journal on Scientific Computing 数学-应用数学

CiteScore

5.50

自引率

3.20%

发文量

209

审稿时长

1 months

期刊介绍： The purpose of SIAM Journal on Scientific Computing (SISC) is to advance computational methods for solving scientific and engineering problems. SISC papers are classified into three categories: 1. Methods and Algorithms for Scientific Computing: Papers in this category may include theoretical analysis, provided that the relevance to applications in science and engineering is demonstrated. They should contain meaningful computational results and theoretical results or strong heuristics supporting the performance of new algorithms. 2. Computational Methods in Science and Engineering: Papers in this section will typically describe novel methodologies for solving a specific problem in computational science or engineering. They should contain enough information about the application to orient other computational scientists but should omit details of interest mainly to the applications specialist. 3. Software and High-Performance Computing: Papers in this category should concern the novel design and development of computational methods and high-quality software, parallel algorithms, high-performance computing issues, new architectures, data analysis, or visualization. The primary focus should be on computational methods that have potentially large impact for an important class of scientific or engineering problems.