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

Author

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  • Yang, Suxiang
  • Chen, Huanzhen
  • Ervin, Vincent J.
  • Wang, Hong

Abstract

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<θ<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.

Suggested Citation

  • Yang, Suxiang & Chen, Huanzhen & Ervin, Vincent J. & Wang, Hong, 2021. "Solvability and approximation of two-side conservative fractional diffusion problems with variable-Coefficient based on least-Squares," Applied Mathematics and Computation, Elsevier, vol. 406(C).
    • Handle: RePEc:eee:apmaco:v:406:y:2021:i:c:s0096300321003192
    DOI: 10.1016/j.amc.2021.126229
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