This paper aims to construct a two-grid mixed finite element scheme for distributed optimal control governed by semilinear elliptic equations. The state and co-state are approximated by the $ P_0^2 $-$ P_1 $ pair and the control variable is approximated by the piecewise constant functions. First, a superclose result for the control variable and a priori error estimates for all variables are obtained. Second, a two-grid $ P_0^2 $-$ P_1 $ mixed finite element algorithm is presented and the corresponding error is analyzed. In the two-grid scheme, the solution of the semilinear elliptic optimal control problem on a fine grid is reduced to the solution of the semilinear elliptic optimal control problem on a much coarser grid and the solution of a linear decoupled algebraic system on the fine grid and the resulting solution still maintains an asymptotically optimal accuracy. We find that the two-grid method achieves the same convergence property as the $ P_0^2 $-$ P_1 $ mixed finite element method if the two mesh sizes satisfy $ h = H^2 $. Finally, a numerical example demonstrating our theoretical results is presented.

In this paper, a two-grid finite element scheme for semilinear parabolic integro-differential equations is proposed. In the two-grid scheme, continuous linear element is used for spatial discretization, while Crank-Nicolson scheme and Leap-Frog scheme are ultilized for temporal discretization. Based on the combination of the interpolation and Ritz projection technique, some superclose estimates between the interpolation and the numerical solution in the $ H^1 $-norm are derived. Notice that we only need to solve nonlinear problem once in the two-grid scheme, namely, the first time step on the coarse-grid space. A numerical example is presented to verify the effectiveness of the proposed two-grid scheme.

In this paper we discuss a priori error estimates and superconvergence of splitting positive definite mixed finite element methods for optimal control problems governed by pseudo-hyperbolic integro-differential equations. The state variables and co-state variables are approximated by the lowest order Raviart–Thomas mixed finite element functions, and the control variable is approximated by piecewise constant functions. First, we derive a priori error estimates for the control variable, state variables, and co-state variables. Second, we obtain a superconvergence result for the control variable.

This paper aims to construct a two-grid scheme for distributed optimal control governed by semilinear elliptic equations. The state and co-state are approximated by the P_0^2-P_1 pair and the control variable is approximated by piecewise constant functions. First, a priori error estimates for the control variable, the state variables and the co-state variables are obtained. Second, a two-grid P_0^2-P_1 mixed finite element scheme is presented and the corresponding error is analyzed. Finally, a numerical example demonstrating our theoretical results is also presented in this paper.

In this paper, we construct a new linear second-order finite difference scheme with two parameters for space-fractional Allen–Cahn equations. We first prove that the discrete maximum principle holds under reasonable constraints on time step size and coefficient of stabilized term. Secondly, we analyze the maximum-norm error. Thirdly, we can see that the proposed scheme is unconditionally energy-stable by defining the modified energy and selecting the appropriate parameters. Finally, two numerical examples are presented to verify the theoretical results.

In this paper, we will give a new Crank–Nicolson mixed covolume method for parabolic optimal control problems. The state and costate variables are approximated by the lowest order Raviart–Thomas element and the control variable is approximated by piecewise constant function, while Crank–Nicolson scheme is ultilized for temporal discretization. We derive the priori error estimates for the control variable, the state and the costate variables.