On nonnegativity preservation in finite element methods for subdiffusion equations
Journal article, 2017

We consider three types of subdiffusion models, namely singleterm, 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.

Nonnegativity preservation


Caputo fractional derivative

Finite element method


Bangti Jin

University College London (UCL)

R. D. Lazarov

Texas A&M University

Vidar Thomee

Chalmers, Mathematical Sciences

University of Gothenburg

Zhi Zhou

Columbia University

Mathematics of Computation

0025-5718 (ISSN) 1088-6842 (eISSN)

Vol. 86 307 2239-2260

Subject Categories

Computational Mathematics

Mathematical Analysis


Basic sciences



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