Two-Level Error Estimation for the Integral Fractional Laplacian

IF 1 4区数学 Q3 MATHEMATICS, APPLIED

Computational Methods in Applied Mathematics Pub Date : 2023-02-14 DOI:10.1515/cmam-2022-0195

Markus Faustmann, Ernst P. Stephan, David Wörgötter

引用次数: 0

Abstract

Abstract For the singular integral definition of the fractional Laplacian, we consider an adaptive finite element method steered by two-level error indicators. For this algorithm, we show linear convergence in two and three space dimensions as well as convergence of the algorithm with optimal algebraic rates in 2D, when newest vertex bisection is employed for mesh refinement. A key step hereby is an equivalence of the nodal and Scott–Zhang interpolation operators in certain weighted L 2 L^{2} -norms.

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积分分数阶拉普拉斯算子的两级误差估计

摘要针对分数阶拉普拉斯函数的奇异积分定义，考虑了一种由两级误差指标引导的自适应有限元方法。对于该算法，我们证明了在二维和三维空间上的线性收敛性，以及在二维空间上采用最新顶点平分进行网格细化时算法的最优代数速率的收敛性。这里的一个关键步骤是在某些加权的l2 L^{2}范数中节点和Scott-Zhang插值算子的等价性。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Computational Methods in Applied Mathematics MATHEMATICS, APPLIED-

CiteScore

2.40

自引率

7.70%

发文量

期刊介绍： The highly selective international mathematical journal Computational Methods in Applied Mathematics　(CMAM) considers original mathematical contributions to computational methods and numerical analysis with applications mainly related to PDEs. CMAM seeks to be interdisciplinary while retaining the common thread of numerical analysis, it is intended to be readily readable and meant for a wide circle of researchers in applied mathematics. The journal is published by De Gruyter on behalf of the Institute of Mathematics of the National Academy of Science of Belarus.