Numerical methods applicable to the approximation of spectral fractional diffusion operators in multidimensional domains with general geometry are analyzed. Over the past decade, several approaches have been proposed to approximate the inverse operator \(\mathcal {A}^{-\alpha }\), \(\alpha \in (0,1)\). Despite their different origins, they can all be written as a rational approximation. Let the matrix \(\mathbb {A}\) be obtained after finite difference or finite element discretization of \(\mathcal {A}\). The BURA (Best Uniform Rational Approximation) method was introduced to approximate the inverse matrix \({\mathbb A}^{-\alpha }\) based on an approximation of the scallar function \(z^\alpha \), \(\alpha \in (0,1)\), \(z\in [0,1]\). In this paper we study BURA and BURA-based methods for fractional powers of sparse symmetric and positive definite (SPD) matrices, presentiing the concept, general framework and error analysis. Our contributions concern approximations of \(\mathbb {A}^{-\alpha }\) and \(\mathbb {A}^\alpha \) for arbitrary \(\alpha > 0\), thus significantly expanding the range of available currently results. Assymptotically accurate error estimates are obtained. The rate of convergence is exponential with respect to the degree of BURA. Numerical results are presented to illustrate and better interpret the theoretical estimates.

分析了适用于具有一般几何的多维域中谱分数扩散算子的近似的数值方法。在过去的十年中，已经提出了几种方法来近似逆算子\(\mathcal {A}^{-\alpha }\)、\(\alpha \in (0,1)\)。尽管它们的起源不同，但它们都可以写成有理近似。令\(\mathcal {A}\)经有限差分或有限元离散后得到矩阵\(\mathbb {A}\)。引入 BURA（最佳均匀有理数逼近）方法，基于标量函数\(z^\alpha \)的近似来逼近逆矩阵\({\mathbb A}^{-\alpha }\)、\( \alpha \in (0,1)\)、\(z\in [0,1]\)。在本文中，我们研究了稀疏对称和正定（SPD）矩阵的分数幂的 BURA 和基于 BURA 的方法，提出了概念、一般框架和误差分析。我们的贡献涉及任意\(\alpha > 0\)的\(\mathbb {A}^{-\alpha }\)和\(\mathbb {A}^\alpha \)的近似值，从而显着扩展了目前可获得的结果。获得渐近准确的误差估计。收敛速度与 BURA 程度成指数关系。给出的数值结果是为了说明和更好地解释理论估计。