In this paper, we study diagonal dominance of the stiffness matrix resulted from the piecewise linear finite element discretisation of the integral fractional Laplacian under global homogeneous Dirichlet boundary condition in one spatial dimension. We first derive the exact form of this matrix in the frequency space which is extendable to multi-dimensional rectangular elements. Then we give the complete answer when the stiffness matrix can be strictly diagonally dominant. As one application, we apply this notion to the construction of maximum principle preserving schemes for the fractional-in-space Allen–Cahn equation, and provide ample numerical results to verify our findings.

在本文中，我们研究了在整体齐次Dirichlet边界条件下在一维空间中积分分数拉普拉斯算子的分段线性有限元离散化所产生的刚度矩阵的对角优势。我们首先在频率空间中得出该矩阵的精确形式，该形式可以扩展到多维矩形元素。然后，当刚度矩阵严格对角占优时，我们给出完整的答案。作为一种应用，我们将此概念应用于空间分数Allen-Cahn方程的最大原理保存方案的构造，并提供足够的数值结果来验证我们的发现。