Local analysis of an L1/finite element method for a time-fractional singularly perturbed reaction-diffusion problem

Meng, Xiangyun ; Gracia, José Luis (Universidad de Zaragoza) ; Stynes, Martin

Resumen: An initial-boundary value problem of the form is considered on the space-time domain , with Dirichlet boundary and initial conditions, where is a Caputo fractional derivative of order and the singular perturbation parameter is a positive constant. Bounds on the solution u and its derivatives are proved by means of a comparison principle with a careful selection of barrier functions; it is seen that u has a weak singularity at the initial time (caused by the fractional derivative) and also has layers (caused by the small parameter ) at the sides of the space-time domain. The Caputo derivative is discretised by the L1 scheme on a graded temporal mesh, then at each time level the PDE is discretised by a piecewise linear finite element method on a Shishkin spatial mesh. Using our bounds on the derivatives of u, error estimates for the computed solution are derived in , energy and balanced norms on [0, 1] for each t; these estimates are local in time and uniform in . Numerical experiments show the sharpness of our theoretical results.

Idioma: Inglés

DOI: 10.1007/s11075-025-02247-x

Año: 2025

Publicado en: NUMERICAL ALGORITHMS (2025), [36 pp.]
ISSN: 1017-1398

Financiación: info:eu-repo/grantAgreement/ES/DGA/E24-23R
Financiación: info:eu-repo/grantAgreement/ES/MICINN/PID2022-141385NB-I00

Tipo y forma: Artículo (Versión definitiva)

Área (Departamento): Área Matemática Aplicada (Dpto. Matemática Aplicada)