We present a finite element scheme for fractional diffusion problems with varying diffusivity and fractional order. We consider a symmetric integral form of these nonlocal equations defined on general geometries and in arbitrary bounded domains. A number of challenges are encountered when discretizing these equations. The first comes from the heterogeneous kernel singularity in the fractional integral operator. The second comes from the dense discrete operator with its quadratic growth in memory footprint and arithmetic operations. An additional challenge comes from the need to handle volume conditions-the generalization of classical local boundary conditions to the nonlocal setting. Satisfying these conditions requires that the effect of the whole domain, including both the interior and exterior regions, can be computed on every interior point in the discretization. Performed directly, this would result in quadratic complexity. To address these challenges, we propose a strategy that decomposes the stiffness matrix into three components. The first is a sparse matrix that handles the singular near-field separately and is computed by adapting singular quadrature techniques available for the homogeneous case to the case of spatially variable order. The second component handles the remaining smooth part of the near-field as well as the far field and is approximated by a hierarchical $\mathcal{H}^{2}$ matrix that maintains linear complexity in storage and operations. The third component handles the effect of the global mesh at every node and is written as a weighted mass matrix whose density is computed by a fast-multipole type method. The resulting algorithm has therefore overall linear space and time complexity. Analysis of the consistency of the stiffness matrix is provided and numerical experiments are conducted to illustrate the convergence and performance of the proposed algorithm.

翻译：我们提出了一种适用于可变扩散系数和分数阶的分数阶扩散问题的有限元格式。我们考虑了这些非局部方程在一般几何形状和任意有界域上的对称积分形式。在离散化这些方程时会遇到若干挑战。第一个挑战来自分数阶积分算子中的异质核奇异性。第二个挑战来自稠密离散算子及其在存储量和算术运算量上的二次增长。另一个挑战来自处理体积条件——即经典局部边界条件在非局部环境中的推广。满足这些条件需要能够计算整个域（包括内部和外部区域）对离散化中每个内部点的影响。若直接执行，这将导致二次复杂度。为应对这些挑战，我们提出了一种将刚度矩阵分解为三个分量的策略。第一个分量是稀疏矩阵，它单独处理奇异近场，通过将适用于齐次情况的奇异求积技术扩展到空间变阶情况进行计算。第二个分量处理剩余的光滑近场以及远场，并通过分层$\mathcal{H}^{2}$矩阵进行近似，该矩阵在存储和运算中保持线性复杂度。第三个分量处理每个节点处的全局网格效应，并写成加权质量矩阵，其密度通过快速多极类方法计算。因此，所得到的算法具有整体线性的空间和时间复杂度。我们还提供了刚度矩阵一致性的分析，并通过数值实验说明了所提出算法的收敛性和性能。