[论文解读] $h-p$ spectral element methods for three dimensional elliptic problems on non-smooth domains using parallel computers

Akhlaq Husain|arXiv (Cornell University)|Oct 11, 2011

Advanced Numerical Methods in Computational Mathematics参考文献 37被引用 9

一句话总结

本文提出了一种非协调 h-p 谱有限元方法，用于在具有指数精度的非光滑区域上求解三维椭圆边值问题。通过在奇点附近采用局部球坐标和柱坐标，并使用在角落和边缘处逐渐加密的几何网格，该方法通过预处理共轭梯度法（PCGM）求解的最小二乘配点法实现最优收敛速率，避免了显式组装质量矩阵和刚度矩阵。

ABSTRACT

Elliptic partial differential equations arise in many fields of science and engineering such as steady state distribution of heat, fluid dynamics, structural/mechanical engineering, aerospace engineering and seismology etc. In three dimensions it is well known that the solutions of elliptic boundary value problems have singular behavior near the corners and edges of the domain. The singularities which arise are known as vertex, edge, and vertex-edge singularities. We propose a nonconforming h-p spectral element method to solve three dimensional elliptic boundary value problems on non-smooth domains to exponential accuracy. To overcome the singularities which arise in the neighbourhoods of the vertices, vertex-edges and edges we use local systems of coordinates. These local coordinates are modified versions of spherical and cylindrical coordinate systems in their respective neighbourhoods. Away from these neighbourhoods standard Cartesian coordinates are used. In each of these neighbourhoods we use a geometrical mesh which becomes finer near the corners and edges. We then derive differentiability estimates in these new set of variables and a stability estimate on which our method is based for a non-conforming h-p spectral element method. The Sobolev spaces in vertex-edge and edge neighbourhoods are anisotropic and become singular at the corners and edges. The method is essentially a least-squares collocation} method and a solution can be obtained using Preconditioned Conjugate Gradient Method (PCGM). To solve the minimization problem we need to solve the normal equations for the least-squares problem. The residuals in the normal equations can be obtained without computing and storing mass and stiffness matrices. Computational results for a number of model problems confirm the theoretical estimates obtained for the error and computational complexity.

研究动机与目标

• 解决三维椭圆 PDE 解在角落、棱边和顶点-棱边区域附近的奇异性行为。
• 开发一种高阶数值方法，即使在解存在奇异性的情况下也能实现指数收敛。
• 通过非协调谱有限元格式，实现大规模问题的高效并行计算。
• 在求解过程中避免组装和存储质量矩阵与刚度矩阵的计算开销。
• 通过模型问题的数值实验验证理论误差估计和计算复杂度。

提出的方法

• 该方法在奇点区域（顶点、棱边、顶点-棱边）附近使用局部坐标系——改进的球坐标和柱坐标——以更精确地解析奇异性。
• 在这些局部邻域中采用几何网格，向奇点处逐渐细化，以捕捉解的各向异性行为。
• 在远离奇点的区域使用标准笛卡尔坐标，确保与标准谱有限元方法的兼容性。
• 该公式基于最小二乘配点法，通过弱形式最小化残差，无需显式组装矩阵。
• 在新坐标系中推导出稳定性和可微性估计，特别是在反映奇异性结构的各向异性 Sobolev 空间中。
• 所得到的线性系统通过预处理共轭梯度法（PCGM）求解，残差直接从法方程计算，无需存储质量或刚度矩阵。

实验结果

研究问题

• RQ1 h-p 谱有限元方法能否在具有角落和棱边奇异性的三维非光滑区域上实现椭圆问题的指数收敛？
• RQ2 局部坐标变换如何改善三维区域中奇异性解的分辨率？
• RQ3 在奇异性附近采用各向异性网格加密对收敛速率和计算成本有何影响？
• RQ4 最小二乘配点法能否在不组装和存储大规模质量与刚度矩阵的情况下高效实现？
• RQ5 理论误差估计在实际大规模计算中与数值结果的对比如何？

主要发现

• 该方法在 h-p 意义下实现了指数收敛速率，验证了针对顶点、棱边和顶点-棱边奇异性问题的理论预测。
• 使用局部球坐标和柱坐标显著提高了在几何奇异性附近对奇异性解的分辨率。
• 计算结果表明收敛行为与推导出的理论误差估计一致，表现出最优收敛性。
• 该方法避免了质量矩阵和刚度矩阵的存储与组装，降低了内存开销，并提升了在并行架构上的可扩展性。
• 预处理共轭梯度法（PCGM）能有效求解由最小二乘公式产生的大规模线性系统。
• 对模型问题的数值实验验证了理论误差界和计算复杂度估计，表明在各种奇异性构型下均具有鲁棒性能。