The ultraspherical spectral element method

We introduce a novel spectral element method based on the ultraspherical and hierarchical Poincar\'{e}-Steklov scheme for solving second-order linear partial differential equations polygonal domains with unstructured quadrilateral or triangular meshes. Properties of lead to almost banded systems, allowing be competitive in high-polynomial regime ($p > 5$). The enables precomputed solution operators reused, fast elliptic solves implicit semi-implicit time-steppers. resulting achieves an overall computational complexity $\mathcal{O}(p^4/h^3)$ mesh size $h$ polynomial order $p$, enabling $hp$-adaptivity efficiently performed. develop open-source software system, ultraSEM, flexible, user-friendly computations MATLAB.