%0 Journal Article
%@ 1088-6842
%A Jin, B
%A Lazarov, R
%A Thomee, V
%A Zhou, Z
%D 2017
%F discovery:1477538
%J Mathematics of Computation
%K subdiffusion, finite element method, nonnegativity preservation, Caputo derivative
%N 307
%P 2239-2260
%T On nonnegativity preservation in finite element methods for subdiffusion equations
%U https://discovery.ucl.ac.uk/id/eprint/1477538/
%V 86
%X 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.
%Z © Copyright 2016 American Mathematical Society. This version is the author accepted manuscript. For information on re-use, please refer to the publisher’s terms and conditions.