We introduce a novel spectral element method based on the ultraspherical spectral method and the hierarchical Poincar\'{e}-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 Poincar\'{e}-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.

翻译：我们采用了一种基于超球光谱法和分级Poincar\'{e}-Steklov办法的新型光谱元素方法,用于解决具有无结构的四边形或三角环形多边形域的二级线性线性局部方程式。超球光谱方法的属性导致几乎带宽线性系统,使该元素方法在高球系体系中具有竞争力(Pp > 5美元)。Poincar\'{e}-Steklov办法使预先计算的解决办法操作者能够被再利用,从而允许在隐含和半隐含时间步器中快速流离解部分方程式。由此形成的光谱元素方法在MATLAB中实现一个以毫什大小为$\mathcal{O}(pä4/h ⁇ 3美元)和多球级为$p$的计算复杂度,使美元能够有效运行。我们开发了一个开放源软件系统,即超SEM,用于灵活、用户友好的光谱元素计算。