In this paper, we consider approximations of Neumann problems for the integral fractional Laplacian by continuous, piecewise linear finite elements. We analyze the weak formulation of such problems, including their well-posedness and asymptotic behavior of solutions. We address the convergence of the finite element discretizations and discuss the implementation of the method. Finally, we present several numerical experiments in one- and two-dimensional domains that illustrate the method’s performance as well as certain properties of solutions.

中文翻译：

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分数诺依曼问题的有限元逼近

在本文中，我们考虑通过连续的分段线性有限元对积分分数拉普拉斯算子的 Neumann 问题进行逼近。我们分析了此类问题的弱表述，包括它们的适定性和解决方案的渐近行为。我们解决了有限元离散化的收敛性并讨论了该方法的实现。最后，我们在一维和二维域中展示了几个数值实验，说明了该方法的性能以及解决方案的某些属性。