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 O(N5/2), where N is the problem size. Numerical examples 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.

中文翻译：

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分数拉普拉斯算子的 hp-FEM 的实现


我们考虑 hp 有限元对 1d 积分狄利克雷分数拉普拉斯算子的离散化。我们提出了正交方案来设置刚度矩阵和载荷矢量，以保持 hp-FEM 在几何网格上的指数收敛。这些方案基于 Gauss-Jacobi 和 Gauss-Legendre 规则。我们表明，取一些略高于多项式次数的正交点就足以保持根指数收敛性。设置系统的代数运算总数为 O（N5/2），其中 N 是问题大小。数值示例说明了分析。我们还将分析扩展到基于形状规则网格的 hp 有限元空间的更高维度的分数阶拉普拉斯算子。