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arXiv · 2202.13357

A posteriori error analysis for variable-coefficient multiterm time-fractional subdiffusion equations

Natalia Kopteva·Martin Stynes
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Abstract

An initial-boundary value problem of subdiffusion type is considered; the temporal component of the differential operator has the form $\sum_{i=1}^{\ell}q_i(t)\, D _t ^{α_i} u(x,t)$, where the $q_i$ are continuous functions, each $D _t ^{α_i}$ is a Caputo derivative, and the $α_i$ lie in $(0,1]$. Maximum/comparison principles for this problem are proved under weak hypotheses. A new positivity result for the multinomial Mittag-Leffler function is derived. A posteriori error bounds are obtained in $L_2(Ω)$ and $L_\infty(Ω)$, where the spatial domain $Ω$ lies in $\bR^d$ with $d\in\{1,2,3\}$. An adaptive algorithm based on this theory is tested extensively and shown to yield accurate numerical solutions on the meshes generated by the algorithm.

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Natalia Kopteva, Martin Stynes. 2022-06-23. A posteriori error analysis for variable-coefficient multiterm time-fractional subdiffusion equations. https://arxiv.org/abs/2202.13357

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