Exponential Convergence of Gauss-Jacobi Quadratures for Singular Integrals over Simplices in Arbitrary Dimension
Exponential Convergence of Gauss-Jacobi Quadratures for Singular Integrals over Simplices in Arbitrary Dimension
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任意维单形上奇异积分的高斯-雅可比求积的指数收敛性
DOI：
10.1137/100812574
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发表时间：
2012
期刊：
影响因子：
--
通讯作者：
C. Schwab
中科院分区：
文献类型：
--
作者：
A. Chernov;C. Schwab
$\mathbb{R}^{d}$ 中积分算子的伽辽金离散化需要计算积分 $\int_{S^{(1)}}\!\int_{S^{(2)}}\!f(x,y)\,dydx$，其中 $S^{(1)},S^{(2)}$ 是 $d$ 维单纯形，$f$ 在 $x=y$ 处有奇点。在[A. Chernov、T. von Petersdorff 和 C. Schwab，M$2$AN 数学。模型。数字。 Anal., 45 (2011), pp. 387--422] 我们构建了一系列 $hp$ 求积规则 ${Q}_N$，对一类被积函数 $f$ 进行 $N$ 函数计算，允许在 $x=y$ 处存在代数奇点，对于 $dx$ 或 $dy$（超奇异核）可能不可积，并且 Gevrey-$\delta$ 平滑$x\ne y$。这对于来自广泛的伪微分算子类别的内核来说是满足的。我们证明$Q_N$达到指数收敛速度$\mathcal{O}(\exp(-rN^\gamma))$，指数为$\gamma = 1/(2d\delta+1)$。在本文中，我们考虑了应用中经常出现的特殊奇点 $\|x-y\|^\alpha$ 和真实的 $\alpha$，并证明了改进的收敛性...
Galerkin discretizations of integral operators in $\mathbb{R}^{d}$ require the evaluation of integrals $\int_{S^{(1)}}\!\int_{S^{(2)}}\!f(x,y)\,dydx$, where $S^{(1)},S^{(2)}$ are $d$-dimensional simplices and $f$ has a singularity at $x=y$. In [A. Chernov, T. von Petersdorff, and C. Schwab, M$2$AN Math. Model. Numer. Anal., 45 (2011), pp. 387--422] we constructed a family of $hp$-quadrature rules ${Q}_N$ with $N$ function evaluations for a class of integrands $f$ allowing for algebraic singularities at $x=y$, possibly nonintegrable with respect to either $dx$ or $dy$ (hypersingular kernels) and Gevrey-$\delta$ smooth for $x\ne y$. This is satisfied for kernels from broad classes of pseudodifferential operators. We proved that $Q_N$ achieves the exponential convergence rate $\mathcal{O}(\exp(-rN^\gamma))$ with the exponent $\gamma = 1/(2d\delta+1)$. In this paper we consider a special singularity $\|x-y\|^\alpha$ with real $\alpha$ which appears frequently in appplication and prove that an improved converg...