Time fractional gradient flows: Theory and numerics
Time fractional gradient flows: Theory and numerics
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DOI：
10.1142/s0218202523500100
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发表时间：
2021-01
期刊：
影响因子：
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通讯作者：
Wenbo Li;A. Salgado
中科院分区：
文献类型：
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作者：
Wenbo Li;A. Salgado
我们开发的分数梯度流理论：一个旨在最小化凸，l.s.c.的演变。能量，具有记忆效应。这种记忆的特征在于，能量的（次）梯度的负值等于状态的所谓卡普托导数。我们引入的概念，能源解决方案，我们提供的存在性，唯一性和一定的正则化效果。我们还考虑了这种能量的Lipschitz扰动。对于这些问题，我们提供了一个后验误差估计，并显示其可靠性。该估计仅依赖于问题数据，并且在连续的时间步长之间不施加约束。在此估计的基础上，我们提供了一个先验的误差分析，使没有假设的解决方案的平滑度。
We develop the theory of fractional gradient flows: an evolution aimed at the minimization of a convex, l.s.c. energy, with memory effects. This memory is characterized by the fact that the negative of the (sub)gradient of the energy equals the so-called Caputo derivative of the state. We introduce the notion of energy solutions, for which we provide existence, uniqueness and certain regularizing effects. We also consider Lipschitz perturbations of this energy. For these problems we provide an a posteriori error estimate and show its reliability. This estimate depends only on the problem data, and imposes no constraints between consecutive time-steps. On the basis of this estimate we provide an a priori error analysis that makes no assumptions on the smoothness of the solution.