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On nonnegativity preservation in finite element methods for subdiffusion equations

Bangti Jin ; Raytcho Lazarov ; Vidar Thomée (Institutionen för matematiska vetenskaper) ; Zhi Zhou
Mathematics of Computation (00255718). Vol. 86 (2017), 307, p. 2239-2260.
[Artikel, refereegranskad vetenskaplig]

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.

Nyckelord: Caputo fractional derivative; Finite element method; Nonnegativity preservation; Subdiffusion

Denna post skapades 2017-06-14. Senast ändrad 2017-07-04.
CPL Pubid: 249778


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