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：本文针对具有可变扩散系数和变分数阶的分数阶扩散问题，提出了一种有限元格式。我们考虑这些非局部方程在一般几何区域和有界域上的对称积分形式。离散这些方程时面临若干挑战：其一来自分数阶积分算子的异质核奇异性；其二来自稠密离散算子及其平方增长的存储与计算开销；此外还需处理体积条件——即经典局部边界条件在非局部框架下的推广。满足这些条件需要计算全区域（包括内部和外部区域）对离散化中每个内部点的作用，直接计算将导致平方复杂度。为应对这些挑战，我们提出将刚度矩阵分解为三个分量的策略：第一个分量为稀疏矩阵，通过将均匀情形下的奇异积分技术扩展到空间变阶情形，单独处理近场奇异部分；第二个分量处理剩余光滑近场及远场，采用分层$\mathcal{H}^{2}$矩阵近似实现存储和运算的线性复杂度；第三个分量处理全局网格对每个节点的作用，通过快速多极子方法计算密度并表示为加权质量矩阵。最终算法整体具有线性空间和时间复杂度。本文还给出了刚度矩阵相容性分析，并通过数值实验验证了所提算法的收敛性和性能。