A Crank–Nicolson Finite Volume Element Method for Time Fractional Sobolev Equations on Triangular Grids

Abstract

In this paper, a finite volume element (FVE) method is proposed for the time fractional Sobolev equations with the Caputo time fractional derivative. Based on the L1-formula and the Crank–Nicolson scheme, a fully discrete Crank–Nicolson FVE scheme is established by using an interpolation operator Ih*. The unconditional stability result and the optimal a priori error estimate in the L2(Ω)-norm for the Crank–Nicolson FVE scheme are obtained by using the direct recursive method. Finally, some numerical results are given to verify the time and space convergence accuracy, and to examine the feasibility and effectiveness for the proposed scheme.

Numerical analysis of time-fractional Sobolev equation for fluid-driven processes in impermeable rocks

“…Haq and Hussain [26,27] developed a meshless scheme based on radial basis functions (RBFs). Beshtokov [28] proposed a finite-difference algorithm, while Qin et al [29] employed a Newton linearized scheme based on the Crank-Nicolson technique and Zhao et al [30] used a finite-volume element (FVE) approach.…”

Vector Arithmetic in the Triangular Grid

“…Although we work with continuous coordinate system which uses real numbers, our work is essential in discrete mathematics, especially, in digital geometry to work, e.g., with digital images on the triangular grid. We should also note that hexagonal, triangular, honeycomb and other related grid structures are used in various other fields, e.g., in networks [ 12 , 13 , 17 , 18 , 19 ], in fractional calculus [ 20 , 21 ], in 3D printing [ 22 ], in chemical and physical modelling [ 23 ] and simulations [ 24 , 25 ], and in city planning [ 26 ], where continuous transformations play also crucial roles, thus our result may be applied. Additionally to the above mentioned fields, triangular grid is applied in skeletonization and thinning algorithms [ 27 , 28 ], in discrete tomography [ 29 ] and in cartography.…”