论文标题
超球频谱元素方法
The ultraspherical spectral element method
论文作者
论文摘要
我们介绍了一种基于超球光谱法和分层庞加尔·斯泰克洛夫方案的新型光谱元素方法，用于求解具有非结构化四边形或三角形网格的多边形域上的二阶线性偏微分方程。超球频谱方法的属性导致几乎带状的线性系统，从而使元素方法在高多项式方案中具有竞争力（$ p> 5 $）。层次的庞加莱 - 斯特克洛夫方案使预算的解决方案操作员可以重复使用，从而可以在隐式和半平整的时间stepper中快速求解。所得的光谱元素方法实现了$ \ Mathcal {o}（p^4/h^3）$的总体计算复杂性，用于网格尺寸$ h $和多项式订单$ p $，启用$ hp $ - 适应性以有效地执行。我们开发了一个开源软件系统Ultrasem，用于MATLAB中的灵活，用户友好的光谱元素计算。
We introduce a novel spectral element method based on the ultraspherical spectral method and the hierarchical Poincaré-Steklov scheme for solving second-order linear partial differential equations on polygonal domains with unstructured quadrilateral or triangular meshes. Properties of the ultraspherical spectral method lead to almost banded linear systems, allowing the element method to be competitive in the high-polynomial regime ($p > 5$). The hierarchical Poincaré-Steklov scheme enables precomputed solution operators to be reused, allowing for fast elliptic solves in implicit and semi-implicit time-steppers. The resulting spectral element method achieves an overall computational complexity of $\mathcal{O}(p^4/h^3)$ for mesh size $h$ and polynomial order $p$, enabling $hp$-adaptivity to be efficiently performed. We develop an open-source software system, ultraSEM, for flexible, user-friendly spectral element computations in MATLAB.
