A convergence result is stated for the numerical solution of space-fractional diffusion problems. For the spatial discretization, an arbitrary family of finite elements can be used combined with the matrix transformation technique. The analysis covers the application of the implicit Euler method for time integration to ensure unconditional stability. The spatial convergence rate does not depend on the fractional power of the Laplacian operator. An efficient numerical implementation is developed avoiding the direct computation of matrix powers.

中文翻译：

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具有最优收敛阶的分数阶扩散问题的有限元方法

给出了空间分数扩散问题数值解的收敛结果。对于空间离散化，可以将任意有限元族与矩阵变换技术结合使用。该分析涵盖了隐式Euler方法在时间积分中的应用，以确保无条件的稳定性。空间收敛率不取决于拉普拉斯算子的分数幂。开发了避免直接计算矩阵幂的有效数值实现。