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In this paper, we study diagonal dominance of the stiffness matrix resultedfrom the piecewise linear finite element discretisation of the integralfractional Laplacian under global homogeneous Dirichlet boundary condition inone spatial dimension. We first derive the exact form of this matrix in thefrequency space which is extendable to multi-dimensional rectangular elements.Then we give the complete answer when the stiffness matrix can be strictlydiagonally dominant. As one application, we apply this notion to theconstruction of maximum principle preserving schemes for thefractional-in-space Allen-Cahn equation, and provide ample numerical results toverify our findings.

On diagonal dominance of FEM stiffness matrix of fractional Laplacian and maximum principle preserving schemes for fractional Allen-Cahn equation