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Abstract In this article, a finite volume element method with the second-order weighted and shifted Grünwald difference (WSGD) formula is proposed and studied for nonlinear time fractional mobile/immobile transport equations on triangular grids. By using the WSGD formula of approximating the Riemann–Liouville fractional derivative and an interpolation operator I h ∗ $I_{h}^{*}$ , a second-order fully discrete finite volume element (FVE) scheme is formulated. The existence, uniqueness, and unconditional stability for the fully discrete FVE scheme are derived, the optimal a priori error estimates in L ∞ ( L 2 ( Ω ) ) $L^{\infty }(L^{2}(\varOmega))$ and L 2 ( H 1 ( Ω ) ) $L^{2}(H^{1}(\varOmega))$ norms are obtained, in which the convergence orders are independent of the fractional parameters. At the end of this article, two numerical examples with different nonlinear terms are given to verify the feasibility and effectiveness.

In this article, a finite volume element method with the second-order weighted and shifted Grünwald difference (WSGD) formula is proposed and studied for nonlinear time fractional mobile/immobile transport equations on triangular grids. By using the WSGD formula of approximating the Riemann–Liouville fractional derivative and an interpolation operator I h ∗ , a second-order fully discrete finite volume element (FVE) scheme is formulated. The existence, uniqueness, and unconditional stability for the fully discrete FVE scheme are derived, the optimal a priori error estimates in L ∞ (L 2 (Ω)) and L 2 (H 1 (Ω)) norms are obtained, in which the convergence orders are independent of the fractional parameters. At the end of this article, two numerical examples with different nonlinear terms are given to verify the feasibility and effectiveness. [ABSTRACT FROM AUTHOR]

Abstract The expanded mixed covolume Element (EMCVE) method is studied for the two-dimensional integro-differential equation of Sobolev type. We use a piecewise constant function space and the lowest order Raviart-Thomas ( RT 0 $\mathit{RT}_{0}$ ) space as the trial function spaces of the scalar unknown u and its gradient σ and flux λ, respectively. The semi-discrete and backward Euler fully-discrete EMCVE schemes are constructed, and the optimal a priori error estimates are derived. Moreover, numerical results are given to verify the theoretical analysis.

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.

The expanded mixed covolume Element (EMCVE) method is studied for the two-dimensional integro-differential equation of Sobolev type. We use a piecewise constant function space and the lowest order Raviart-Thomas ( $\mathit{RT}_{0}$ ) space as the trial function spaces of the scalar unknown u and its gradient σ and flux λ, respectively. The semi-discrete and backward Euler fully-discrete EMCVE schemes are constructed, and the optimal a priori error estimates are derived. Moreover, numerical results are given to verify the theoretical analysis. [ABSTRACT FROM AUTHOR]

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 L 1 -formula and the Crank–Nicolson scheme, a fully discrete Crank–Nicolson FVE scheme is established by using an interpolation operator I h * . The unconditional stability result and the optimal a priori error estimate in the L 2 (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. [ABSTRACT FROM AUTHOR]