We consider the discretization of the $1d$-integral Dirichlet fractional Laplacian by $hp$-finite elements. We present quadrature schemes to set up the stiffness matrix and load vector that preserve the exponential convergence of $hp$-FEM on geometric meshes. The schemes are based on Gauss-Jacobi and Gauss-Legendre rules. We show that taking a number of quadrature points slightly exceeding the polynomial degree is enough to preserve root exponential convergence. The total number of algebraic operations to set up the system is $\mathcal{O}(N^{5/2})$, where $N$ is the problem size. Numerical example illustrate the analysis. We also extend our analysis to the fractional Laplacian in higher dimensions for $hp$-finite element spaces based on shape regular meshes.

翻译：本文研究一维积分狄利克雷分数阶拉普拉斯算子的hp有限元离散方法。我们提出了构建刚度矩阵和载荷向量的数值积分方案，该方案能保持hp有限元在几何网格上指数收敛的特性。该方案基于高斯-雅可比与高斯-勒让德积分法则。研究表明，只需选取略超过多项式次数的积分点数量即可保持根指数收敛特性。构建方程系统的总代数运算量为$\mathcal{O}(N^{5/2})$，其中$N$为问题规模。数值算例验证了理论分析。我们还将该分析推广至高维情形下基于形状正则网格的hp有限元空间中的分数阶拉普拉斯算子。