n维平行六面体中具有尖锐梯度和尖点的函数的有效自适应积分
Efficient adaptive integration of functions with sharp gradients and cusps in n-dimensional parallelepipeds
AI总结：
提出一种基于分治策略的自适应积分算法，用于高效积分具有尖锐梯度和尖点的函数，在Rn中C0函数达到n+1阶收敛速度，并应用于正则化Heaviside函数积分和晶体金刚石全电子库仑问题的富集有限元求解。
摘要：
本文研究了具有尖锐梯度和尖点函数的高效数值积分。提出了一种自适应积分算法，系统性地提高一组函数积分的精度。该算法基于分治策略，且与尖锐梯度或尖点的位置无关。误差分析表明，对于C^0函数（在某点导数不连续），在R^n中可获得n+1的收敛速度。研究了自适应积分方案的两个应用。首先，将自适应求积用于正则化Heaviside函数（一种用于模拟尖锐梯度的强局部化函数）的积分。然后，将自适应求积用于晶体金刚石全电子库仑问题的富集有限元求解。该问题的源项和富集函数在原子核处具有尖锐梯度和尖点。我们表明，与具有相同单元数的纯有限元解相比，仅以积分点数的微小增加即可获得最优收敛速度。该自适应积分方案简单、鲁棒，可直接应用于任何采用具有尖锐局部变化或尖点的富集的广义有限元方法，适用于n维平行六面体单元。
英文摘要：
In this paper, we study the efficient numerical integration of functions with sharp gradients and cusps. An adaptive integration algorithm is presented that systematically improves the accuracy of the integration of a set of functions. The algorithm is based on a divide and conquer strategy and is independent of the location of the sharp gradient or cusp. The error analysis reveals that for a $C^0$ function (derivative-discontinuity at a point), a rate of convergence of $n+1$ is obtained in $R^n$. Two applications of the adaptive integration scheme are studied. First, we use the adaptive quadratures for the integration of the regularized Heaviside function---a strongly localized function that is used for modeling sharp gradients. Then, the adaptive quadratures are employed in the enriched finite element solution of the all-electron Coulomb problem in crystalline diamond. The source term and enrichment functions of this problem have sharp gradients and cusps at the nuclei. We show that the optimal rate of convergence is obtained with only a marginal increase in the number of integration points with respect to the pure finite element solution with the same number of elements. The adaptive integration scheme is simple, robust, and directly applicable to any generalized finite element method employing enrichments with sharp local variations or cusps in $n$-dimensional parallelepiped elements.
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