Optimal $L^\infty(L^2)$ and $L^1(L^2)$ a posteriori error estimates for the fully discrete approximations of time fractional parabolic differential equations

Jiliang Cao; Wansheng Wang; Aiguo Xiao

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arXiv · 2311.06711

Optimal $L^\infty(L^2)$ and $L^1(L^2)$ a posteriori error estimates for the fully discrete approximations of time fractional parabolic differential equations

Jiliang Cao·Wansheng Wang·Aiguo Xiao

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Abstract

We derive optimal order a posteriori error estimates in the $L^\infty(L^2)$ and $L^1(L^2)$-norms for the fully discrete approximations of time fractional parabolic differential equations. For the discretization in time, we use the $L1$ methods, while for the spatial discretization, we use standard conforming finite element methods. The linear and quadratic space-time reconstructions are introduced, which are generalizations of the elliptic space reconstruction. Then the related a posteriori error estimates for the linear and quadratic space-time reconstructions play key roles in deriving global and pointwise final error estimates. Numerical experiments verify and complement our theoretical results.

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Jiliang Cao, Wansheng Wang, Aiguo Xiao. 2023-11-12. Optimal $L^\infty(L^2)$ and $L^1(L^2)$ a posteriori error estimates for the fully discrete approximations of time fractional parabolic differential equations. https://arxiv.org/abs/2311.06711

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