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

TL;DR

This paper investigates the diagonal dominance of FEM stiffness matrices for the fractional Laplacian and develops maximum principle preserving schemes for the fractional Allen-Cahn equation, supported by numerical verification.

Contribution

It provides a detailed analysis of diagonal dominance conditions for FEM matrices of the fractional Laplacian and applies this to construct stable schemes for the fractional Allen-Cahn equation.

Findings

Derived the exact form of the stiffness matrix in frequency space

Identified conditions for strict diagonal dominance of the matrix

Validated the schemes through numerical experiments

Abstract

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.