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In this paper, we present a two-grid lumped mass finite element algorithm for 2D Allen-Cahn equations, where Crank-Nicolson scheme and piecewise linear element are utilized for temporal and spatial discretization, respectively. Both the maximum-norm boundedness and H 1 -norm error estimates of the proposed two-grid scheme are discussed. Finally, a numerical example is given to verify the theoretical results. By comparing with the finite element scheme and the two-grid finite element scheme, we found that our scheme has a great advantage in calculation time. [ABSTRACT FROM AUTHOR]

In this paper, we present a two-grid scheme of expanded mixed finite element method combined with two second-order time discretization schemes for semilinear parabolic integro-differential equations and give a detailed convergence analysis. On the coarse grid space, we first use the Crank–Nicolson scheme to solve the original nonlinear problem at the first time step, then we utilize the Leap-Frog scheme at the rest time levels. Next, we make use of the known coarse mesh solution and Taylor expansion to infer the solution on the fine mesh space. Thus, we only need to solve the nonlinear problem once on the coarse grid space of the two-grid scheme. Finally, a numerical example is presented to verify the effectiveness of the proposed two-grid scheme. [ABSTRACT FROM AUTHOR]

In this paper, we present a two‐grid mixed finite element scheme for distributed optimal control governed by general elliptic equations. P02–P1 mixed finite elements are used for the discretization of the state and co‐state variables, whereas piecewise constant function is used to approximate the control variable. We first use a new approach to obtain the superclose property between the centroid interpolation and the numerical solution of the optimal control u with order h2 under the low regularity. Based on the superclose property, we derive the optimal a priori error estimates. Then, using a postprocessing projection operator, we get a second‐order superconvergent result for the control u. Next, we construct a two‐grid mixed finite element scheme and analyze a priori error estimates. In the two‐grid scheme, the solution of the elliptic optimal control problem on a fine grid is reduced to the solution of the elliptic optimal control problem on a much coarser grid and the solution of a linear algebraic system on the fine grid and the resulting solution still maintains an asymptotically optimal accuracy. Finally, a numerical example is presented to verify the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we discuss a priori error estimates of two-grid mixed finite element methods for a class of nonlinear parabolic equations. The lowest order Raviart-Thomas mixed finite element and Crank-Nicolson scheme are used for the spatial and temporal discretization. First, we derive the optimal a priori error estimates for all variables. Second, we present a two-grid scheme and analyze its convergence. It is shown that if the two mesh sizes satisfy h = H2, then the two-grid method achieves the same convergence property as the Raviart-Thomas mixed finite element method. Finally, we give a numerical example to verify the theoretical results. [ABSTRACT FROM AUTHOR]

For shaped charge blasting projects in mining, civil engineering, and similar fields, it is proposed to modify the charge structure by combining slotted tubes and shaped charge liners to obtain a new type of charge structure. This aims to achieve directional rock breaking through the focused action of the shaped charge. The influence of different slotted pipe materials on the directional rock-breaking effect of concentrated energy using a new charge structure is explored through theoretical analysis combined with model test study, high-speed camera, stress–strain gauge, and other equipment. A comparison is made between slotted pipes made of aluminum, kraft paper, and PVC, with the cutting width of 2 mm. Based on the characteristics of the cracks formed after blasting, the new charge structure made of aluminum slotted pipe produces a penetrating crack that is almost consistent with the pre-cracking direction. Based on the corresponding characteristics of successively released blasting energy, the guiding and convergence effect of the new charge structure made of aluminum slotted pipe on the explosion energy is greater than that of the new charge structure made of the other two types of slotted pipe material. According to the strain data measured after blasting, the peak arrival time of the strain peak in the direction of the slotted pipe on one side of the shaped hood is shorter than that in the other two directions, and the peak strain is greater than that in the other two directions while having a better energy gathering effect. Based on the findings, the new charge structure with directional energy concentration has a damage reduction effect. Furthermore, the material of aluminum slotted pipe is found to be better than PVC slotted pipe, whereas the material of PVC slotted pipe is better than kraft paper slotted pipe in achieving directional rock breaking. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we present a two-grid mixed finite element scheme for distributed optimal control problems governed by stationary Stokes equations. In order to avoid the difficulty caused by the symmetry constraint of the stress tensor, we use pseudostress to replace it. The state and co-state are approximated by the lowest order Raviart–Thomas mixed finite element spaces and the control variable is approximated by piecewise constant functions. We first prove that the difference between the interpolation and the numerical solution has superconvergence property for the control u with order h 2. Then, using the postprocessing technique, we derive a second-order superconvergent result for the control u . Next, we construct a two-grid mixed finite element scheme and derive a priori error estimates. Finally, a numerical experiment is presented to verify the theoretical results. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we discuss a priori error estimates of two-grid mixed finite element methods for a class of nonlinear parabolic equations. The gradient for the method belongs to the square integrable space instead of the classical H (div ; Ω) space. P 0 2 – P 1 mixed finite elements and Crank–Nicolson method are used for the spatial and temporal discretization. First, we derive the optimal a priori error estimates for all variables. Second, we present a two-grid scheme and analyze its convergence. It is shown that when the two mesh sizes satisfy h = H 2 , the two-grid method achieves the same convergence property as the P 0 2 – P 1 mixed finite element method. Finally, we give a numerical example to verify the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a two grid discretization scheme for semilinear parabolic integro-differential equations by expanded mixed finite element methods. The lowest order Raviart–Thomas mixed finite element method and backward Euler method are used for spatial and temporal discretization respectively. Firstly, expanded mixed Ritz–Volterra projection is defined and the related a priori error estimates are proved. Secondly, a superconvergence property of the pressure variable for the fully discretized scheme is obtained. Thirdly, a two-grid scheme is presented to deal with the nonlinear part of the equation and a rigorous convergence analysis is given. It is shown that when the two mesh sizes satisfy h = H 2 , the two grid method achieves the same convergence property as the expanded mixed finite element method. Finally, a numerical experiment is implemented to verify theoretical results of the two grid method. [ABSTRACT FROM AUTHOR]

The shape of a charge liner used in shaped charges with a combined liner will greatly influence the blasting effect. In this study, we examined how combined charge liners with different shapes affected directional rock blasting, and we assessed the influence mechanism. The numerical simulation results showed that among the three shaped charge liners, liners with arc and triangular shapes performed significantly better than the flat‐top liner, and the triangular liner was slightly superior to the arc‐shaped liner. Model testing indicated that principal cracks in the arc‐shaped and triangular liners developed along the combined energy‐gathering direction, while the principal cracks of the flat‐top liner deviated 32° from the axis. Therefore, the directional cracking effect of arc‐shaped and triangular liners was superior to the flat‐top liner. According to the peak strain values on the liners, the peak strain in the combined energy‐gathering direction was greater than that in the slot and nonenergy‐gathering directions, and the order of the peak strain of the three liners followed arc‐shaped liner > triangular liner > flat‐top liner. The directional blasting effect of the triangular liner was slightly better than the arc‐shaped liner. In conclusion, the findings of this study suggested that the best directional rock‐blasting effect was achieved when the shape of the combined charge liner was triangular. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a priori and a posteriori error estimates of H 1 -Galerkin 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 and linear finite element, and the control variable is approximated by piecewise constant functions. Based on two new elliptic projections, we derive a priori error estimates both for the control variable, the state variable and the co-state variable. The related a priori error estimates for the new projections error are also established. Moreover, a posteriori error estimates for all variables are derived via energy method. Such a posteriori error estimates, which are apparently not available in the literature, are an important step towards developing reliable adaptive mixed finite element approximation schemes for the control problem. [ABSTRACT FROM AUTHOR]

In this paper, we discuss a priori error estimates of H 1 -Galerkin mixed finite element methods for optimal control problems governed by parabolic integro-differential equations. The state variables and co-state variables are approximated by the lowest order Raviart–Thomas mixed finite element and linear finite element, and the control variable is approximated by piecewise constant functions. Both semidiscrete and fully discrete schemes are considered. Based on some new elliptic projections, we derive a priori error estimates for the control variable, the state variables and the adjoint state variables. The related a priori error estimates for the new projections error are also established. A numerical example is given to demonstrate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate error estimates and superconvergence of a mixed finite element method for elliptic optimal control problems. The gradient for our method belongs to the square integrable space instead of the classical H ( div ; Ω ) space. The state and co-state are approximated by the P 0 2 – P 1 (velocity–pressure) pair and the control variable is approximated by piecewise constant functions. First, we derive a priori error estimates in H 1 -norm for the state and the co-state scalar functions, a priori error estimates in ( L 2 ) 2 -norm for the state and the co-state vector functions and a priori error estimates in L 2 -norm for the control function. Then, using postprocessing projection operator, we derive a superconvergence result for the control variable. Next, we get a priori error estimates in L 2 -norm for the state and the co-state scalar functions. Finally, a numerical example is given to demonstrate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate superconvergence and a posteriori error estimates of splitting positive definite mixed finite element methods for elliptic optimal control problems. The presented scheme is independent symmetric and positive definite for the state variables and the adjoint state variables. Moreover, the matching relation (i.e., LBB-condition) between the mixed element spaces V h and W h is not necessary, thus, we can choose the approximation spaces more flexibly. In order to derive the superconvergence, we will use classical mixed finite element spaces. The state and co-state are approximated by the lowest order Raviart–Thomas mixed finite element spaces and the control variable is approximated by piecewise constant functions. At first, we derive some superconvergence properties for the control variable, the state variables and the adjoint state variables. Then, using a recovery operator, we obtain a superconvergence result for the control variable. Next, combining energy approach with postprocessing method, we derive a posteriori error estimates for optimal control problems. We will show that the method does not involve the jump residuals which make the analysis simpler. Finally, a numerical example is given to demonstrate the theoretical results on superconvergence. [ABSTRACT FROM AUTHOR]

In this paper, we investigate numerical approximations of H 1 -Galerkin mixed finite element methods for elliptic optimal control problems. The presented scheme is independent symmetric and positive definite for the state variables and the adjoint state variables. Moreover, the matching relation (i.e., LBB-condition) between the mixed element spaces V h and W h is not necessary, thus, we can choose the approximation spaces more flexibly. The state and co-state are approximated by the lowest order Raviart–Thomas mixed finite element spaces and the standard finite element spaces, the control variable is approximated by piecewise constant functions. We derive a priori and a posteriori error estimates for the control variable, the state variables and the adjoint state variables. Finally, some numerical examples are given to demonstrate the theoretical results about a priori error estimates. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the superconvergence property of the numerical solution of a quadratic parabolic optimal control problem by using fully discrete mixed finite element methods. The space discretization of the state variable is done using usual mixed finite elements, whereas the time discretization is based on difference methods. The state and the co-state are approximated by the lowest order Raviart–Thomas mixed finite element spaces and the control variable is approximated by piecewise constant functions. We derive the superconvergence results for the control and the state approximation. Some numerical examples are presented to confirm the theoretical investigations. [ABSTRACT FROM AUTHOR]

In this article, we investigate the L ∞( L2) -error estimates of the semidiscrete expanded mixed finite element methods for quadratic optimal control problems governed by hyperbolic integrodifferential equations. The state and the costate are discretized by the order k Raviart-Thomas mixed finite element spaces, and the control is approximated by piecewise polynomials of order k( k ≥ 0). We derive error estimates for both the state and the control approximation. Numerical experiments are presented to test the theoretical results. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013 [ABSTRACT FROM AUTHOR]

In this article, we analysea posteriorierror estimates of mixed finite element discretizations for linear parabolic equations. The space discretization is done using the order λ ≥ 1 Raviart–Thomas mixed finite elements, whereas the time discretization is based on discontinuous Galerkin (DG) methods (r ≥ 1). Using the duality argument, we derivea posteriori l∞(L2) error estimates for the scalar function, assuming that only the underlying mesh is static. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the L ∞-error estimates of the numerical solutions of linear-quadratic elliptic control problems by using higher order mixed finite element methods. The state and co-state are approximated by the order k Raviart-Thomas mixed finite element spaces and the control variable is approximated by piecewise polynomials of order k (k≥1). Optimal L ∞-error estimates are derived for both the control and the state approximations. These results are seemed to be new in the literature of the mixed finite element methods for optimal control problems. [ABSTRACT FROM AUTHOR]

This paper aims to present a new second-order maximum-principle preserving finite difference scheme for Allen–Cahn equations with periodic boundary conditions. The discrete maximum principle, the maximum-norm error estimate and the discrete energy stability of the proposed scheme are discussed. It is shown that the proposed scheme is unconditionally energy-stable by choosing proper parameters. A numerical experiment is performed to verify the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we consider finite difference method for solving Allen–Cahn equation which contains small perturbation parameters and strong nonlinearity. We use a stabilized second-order Crank–Nicolson/Adams–Bashforth scheme in time and a second-order finite difference approach in space. It is shown that the numerical solutions satisfy discrete maximum principle under reasonable constraints on time step size and coefficient of stability term. Based on the maximum stability, the discrete energy stability is investigated. Two numerical experiments are performed to verify the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new linear second-order finite difference scheme for Allen–Cahn equations. We use a modified Leap-Frog finite difference scheme with stabilized term and a central finite difference scheme for temporal and spatial discretization respectively. It is shown that the discrete maximum principle holds under reasonable constraints on time step size and coefficient of stabilized term. Based on the maximum stability, the maximum-norm error is analyzed. Moreover, we can see that the proposed scheme is unconditionally energy-stable. Finally, a numerical experiment is performed to verify the theoretical results. [ABSTRACT FROM AUTHOR]