We investigate solvability theory and numerical simulation for two-side conservative fractional diffusion equations (CFDE) with a variable-coefficient $a\left(x\right)$. We introduce $\sigma =-aDp$ as an intermediate variable to isolate $a\left(x\right)$ from the nonlocal operator, and then apply the least-squares method to obtain a mixed-type variational formulation. Correspondingly, solution space is split into a regular space and a kernel-dependent space. The solution $p$ and $\sigma$ are then represented as a sum of a regular part and a kernel-dependent singular part. Doing so, a new regularity theory is established, which extends those regularity results for the one side CFDE in [23, 36], and for the fractional Laplace equation corresponding to $\theta =1/2$ in [37, 38], to general CFDE with variable diffusive coefficients and for $0<\theta <1$. Then, we design a kernel-independent least-squares mixed finite element approximation scheme (LSMFE). Theoretical analysis and numerical simulation demonstrate that the LSMFE can capture the singular part of the solution, approximate the solution with optimal-order accuracy, and can be easily implemented.

我们研究了具有变系数的两侧保守分数阶扩散方程（CFDE）的可溶性理论和数值模拟 $\mathrm{一种}（X）$。我们介绍$\sigma =-\mathrm{一种}dp$ 作为隔离的中间变量 $\mathrm{一种}（X）$从非局部算子中得到，然后应用最小二乘法获得混合型变分公式。相应地，解决方案空间分为常规空间和依赖内核的空间。解决方案$p$ 和 $\sigma$然后将表示为常规部分和依赖于内核的奇异部分之和。这样做，建立了新的正则性理论，将[23，36]中一侧CFDE的正则性结果以及对应于分数阶拉普拉斯方程的正则性结果扩展为$\theta =\mathrm{1个}/\mathrm{2个}$ 在[37，38]中，使用具有可变扩散系数的通用CFDE，并且对于 $0<\theta <\mathrm{1个}$。然后，我们设计了一个独立于内核的最小二乘混合有限元逼近方案（LSMFE）。理论分析和数值模拟表明，LSMFE可以捕获解的奇异部分，以最佳阶精度近似求解，并且易于实现。