On Nonnegativity Preservation in Finite Element Methods for Subdiffusion Equations

TL;DR

This paper investigates whether finite element methods preserve the nonnegativity property in subdiffusion equations, revealing conditions under which nonnegativity is maintained or lost, with theoretical analysis and numerical validation.

Contribution

It provides a comprehensive analysis of nonnegativity preservation in finite element discretizations for various subdiffusion models, extending previous heat equation results.

Findings

Nonnegativity may be lost initially but can reappear after a positivity threshold.

Lumped mass method preserves nonnegativity only on Delaunay triangulations.

Numerical experiments support theoretical predictions.

Abstract

We consider three types of subdiffusion models, namely single-term, multi-term and distributed order fractional diffusion equations, for which the maximum-principle holds and which, in particular, preserve nonnegativity. Hence the solution is nonnegative for nonnegative initial data. Following earlier work on the heat equation, our purpose is to study whether this property is inherited by certain spatially semidiscrete and fully discrete piecewise linear finite element methods, including the standard Galerkin method, the lumped mass method and the finite volume element method. It is shown that, as for the heat equation, when the mass matrix is nondiagonal, nonnegativity is not preserved for small time or time-step, but may reappear after a positivity threshold. For the lumped mass method nonnegativity is preserved if and only if the triangulation in the finite element space is of Delaunay type. Numerical experiments illustrate and complement the theoretical results.