Abstract We consider the numerical approximation of the spectral fractional diffusion problem based on the so called Balakrishnan representation. The latter consists of an improper integral approximated via quadratures. At each quadrature point, a reaction–diffusion problem must be approximated and is the method bottle neck. In this work, we propose to reduce the computational cost using a reduced basis strategy allowing for a fast evaluation of the reaction–diffusion problems. The reduced basis does not depend on the fractional power s for 0 < smin ⩽ s ⩽ smax < 1. It is built offline once for all and used online irrespectively of the fractional power. We analyze the reduced basis strategy and show its exponential convergence. The analytical results are illustrated with insightful numerical experiments.

中文翻译：

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谱分数扩散问题解的简化基近似

摘要 我们考虑基于所谓的 Balakrishnan 表示的光谱分数扩散问题的数值近似。后者由通过正交近似的不当积分组成。在每个正交点，必须近似反应扩散问题，这是方法的瓶颈。在这项工作中，我们建议使用简化的基础策略来降低计算成本，从而可以快速评估反应扩散问题。当 0 < smin ⩽ s ⩽ smax < 1 时，缩减基不依赖于分数幂 s。它是一次性离线构建的，并且与分数幂无关地在线使用。我们分析了缩减基础策略并展示了它的指数收敛性。分析结果用富有洞察力的数值实验来说明。