SIAM Journal on Scientific Computing, Volume 43, Issue 2, Page A1362-A1388, January 2021.
The purpose of this paper is to study a Helmholtz problem with a spectral fractional Laplacian, instead of the standard Laplacian. Recently, it has been established that such a fractional Helmholtz problem better captures the underlying behavior in geophysical electromagnetics. We establish the well-posedness and regularity of this problem. We introduce a hybrid spectral-finite element approach to discretize it and show well-posedness of the discrete system. In addition, we derive a priori discretization error estimates. Finally, we introduce an efficient solver that scales as well as the best possible solver for the classical integer-order Helmholtz equation. We conclude with several illustrative examples that confirm our theoretical findings.

中文翻译：

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分数亥姆霍兹方程的快速求解器

SIAM科学计算杂志，第43卷，第2期，第A1362-A1388页，2021年1月。
本文的目的是研究谱分数拉普拉斯算子而不是标准拉普拉斯算子的亥姆霍兹问题。近来，已经确定的是，这种分数亥姆霍兹问题更好地捕获了地球物理电磁学中的基本行为。我们确定此问题的适定性和规律性。我们引入了一种混合频谱有限元方法来离散化它并显示离散系统的适定性。此外，我们推导了先验离散误差估计。最后，我们为经典整数阶Helmholtz方程引入了可缩放的高效解算器以及最佳解算器。我们以几个例证性例子作为结束，这些例子证实了我们的理论发现。