Solvability and approximation of two-side conservative fractional diffusion problems with variable-Coefficient based on least-Squares

•Prove that two-side fractional operator is surjective from solution space to L2.•Solution space is a sum of a regular space and a kernel-dependent space.•Least-squares Galerkin form is obtained and the regularity theorem is established.•A kernel-capture FE scheme is designed and its optimal convergence is proved. We investigate solvability theory and numerical simulation for two-side conservative fractional diffusion equations (CFDE) with a variable-coefficient a(x). We introduce σ=−aDp as an intermediate variable to isolate a(x) 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 σ 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 θ=1/2 in [37, 38], to general CFDE with variable diffusive coefficients and for 0