A PDE APPROACH TO SPACE-TIME FRACTIONAL PARABOLIC PROBLEMS

作者：Nochetto, Ricardo H.^*; Otarola, Enrique^*; Salgado, Abner J.^*

来源：SIAM Journal on Numerical Analysis, 2016, 54(2): 848-873.

DOI：10.1137/14096308X

• fractional derivatives and integrals
• fractional diffusion
• weighted Sobolev spaces
• finite elements
• stability
• anisotropic estimates
• fully discrete methods

We study solution techniques for parabolic equations with fractional diffusion and Caputo fractional time derivative, the latter being discretized and analyzed in a general Hilbert space setting. The spatial fractional diffusion is realized as the Dirichlet-to-Neumann map for a nonuniformly elliptic problem posed on a semi-infinite cylinder in one more spatial dimension. We write our evolution problem as a quasi-stationary elliptic problem with a dynamic boundary condition. We propose and analyze an implicit fully discrete scheme: first-degree tensor product finite elements in space and an implicit finite difference discretization in time. We prove stability and error estimates for this scheme.

• 出版日期2016