Numerical analysis of a positivity-preserving finite element method for fractional Fisher–KPP equation

In this paper, we investigate the numerical analysis of fractional Fisher–KPP equation with Neumann boundary conditions. We rigorously establish key analytical properties of the exact solution, including positivity, boundedness, asymptotic stability, and regularity. A finite element method combined with an L1-implicit–explicit scheme is proposed to solve the equation. Building upon the diagonally positive-definite structure of the mass matrix, it is shown that both the semi-discrete and fully discrete schemes preserve the qualitative properties of the solution, i.e., the numerical solution remains positive for positive initial data, bounded for bounded initial data, and stable for when the exact solution is stable. We further derive the spatial error estimates by exploiting the boundedness and regularity of the exact solution. Our scheme extends effectively to irregular domains while maintaining these properties. Numerical experiments illustrate and complement the theoretical results.