Definite Mixed Finite Element Method Research Articles

Superconvergence of Semidiscrete Splitting Positive Definite Mixed Finite Elements for Hyperbolic Optimal Control Problems

In this paper, we consider semidiscrete splitting positive definite mixed finite element methods for optimal control problems governed by hyperbolic equations with integral constraints. The state and costate are approximated by the lowest order Raviart-Thomas mixed rectangular finite element, and the control is approximated by piecewise constant functions. We derive some convergence and superconvergence results for the control, the state and the adjoint state. A numerical example is provided to demonstrate our theoretical results.

A new two-grid mixed finite element analysis of semi-linear reaction–diffusion equation

A new mixed finite element method is established for solving the semi-linear reaction–diffusion equation based on two-grid discretization. Here, a symmetric positive definite mixed finite element method is constructed for semi-linear problem, and the corresponding error estimates in L2 and Lp-norms are considered. Then, the two-grid techniques with and without one Newton iteration correction are applied for the proposed mixed element procedure to improve the efficiency of computation. The corresponding convergence of the proposed algorithms is analyzed, and the asymptotically optimal approximation is achieved with the relation H=O(h12). Finally, numerical examples are provided to confirm theoretical results.

A new discontinuous Galerkin mixed finite element method for compressible miscible displacement problem

A new combined discontinuous Galerkin method is proposed for compressible miscible displacement problem in porous media. Here, a splitting positive definite mixed finite element method is used for the pressure and Darcy velocity, while an interior penalty discontinuous Galerkin (IPDG) method is used for the transport equation. The stability and convergence of this algorithm are considered, and the optimal a priori error estimate in l∞(L2) for velocity, pressure and concentration are given. Finally we provide some numerical results to confirm our theoretical analysis, and simulate compressible fluid flows through homogeneous and isotropic porous media.

A Priori Error Estimates and Superconvergence of Splitting Positive Definite Mixed Finite Element Methods for Pseudo-Hyperbolic Integro-Differential Optimal Control Problems

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.

A priori error estimates and superconvergence of splitting positive definite mixed finite element methods for pseudo-hyperbolic integro-differential optimal control problems

A priori error estimates and superconvergence of splitting positive definite mixed finite element methods for pseudo-hyperbolic integro-differential optimal control problems

A Priori Error Estimate of Splitting Positive Definite Mixed Finite Element Method for Parabolic Optimal Control Problems

AbstractIn this paper, we propose a splitting positive definite mixed finite element method for the approximation of convex optimal control problems governed by linear parabolic equations, where the primal state variable y and its flux σ are approximated simultaneously. By using the first order necessary and sufficient optimality conditions for the optimization problem, we derive another pair of adjoint state variables z and ω, and also a variational inequality for the control variable u is derived. As we can see the two resulting systems for the unknown state variable y and its flux σ are splitting, and both symmetric and positive definite. Besides, the corresponding adjoint states z and ω are also decoupled, and they both lead to symmetric and positive definite linear systems. We give some a priori error estimates for the discretization of the states, adjoint states and control, where Ladyzhenkaya-Babuska-Brezzi consistency condition is not necessary for the approximation of the state variable y and its flux σ. Finally, numerical experiments are given to show the efficiency and reliability of the splitting positive definite mixed finite element method.

Superconvergence and a posteriori error estimates of splitting positive definite mixed finite element methods for elliptic optimal control problems

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 Vh and Wh 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.

A splitting positive definite mixed finite element method for elliptic optimal control problem

In this paper, we propose a splitting positive definite mixed finite element method (MFEM) for the approximation of convex optimal control problem governed by elliptic equations with control constraints. By selecting the variation functional properly, the presented procedure can be split into two independent, symmetric and positive definite weak formula for the unknown state variable y and for the unknown flux variable σ. It then follows from the first order necessary and sufficient optimality condition, we deduce another two corresponding adjoint state equations for z and w, which are also independent, symmetric and positive definite. Also, a variational inequality for the control variable u is involved. Convergence analysis shows that the method yields the approximate solutions with optimal accuracy in L2(ΩU)-norm for the control u,L2(Ω)-norm for the original state y and adjoint state z,H(div;Ω)-norm for the flux state σ and adjoint state w without requiring the LBB consistency condition. Finally, some numerical examples are presented to confirm our theoretical results.

Splitting positive definite mixed element method for viscoelasticity wave equation

A splitting positive definite mixed finite element method is proposed for second-order viscoelasticity wave equation. The proposed procedure can be split into three independent symmetric positive definite integro-differential sub-system and does not need to solve a coupled system of equations. Error estimates are derived for both semidiscrete and fully discrete schemes. The existence and uniqueness for semidiscrete scheme are proved. Finally, a numerical example is provided to illustrate the efficiency of the method.

Two splitting positive definite mixed finite element methods for parabolic integro-differential equations

Two novel mixed finite element procedures are established for parabolic integro-differential equations, which can be split into two independent symmetric positive definite sub-schemes and do not need to solve a coupled system of equations without requiring the LBB consistency condition. The convergence analysis shows that both methods lead to the optimal order L2(Ω) norm error estimate for u and optimal H(div;Ω) norm error estimate for σ. A numerical example is presented to illustrate the theoretical analysis.

A splitting positive definite mixed finite element method for two classes of integro-differential equations

In this article, we establish a new mixed finite element procedure to solve the second-order hyperbolic and pseudo-hyperbolic integro-differential equations, in which the mixed element system is symmetric positive definite without requiring the LBB consistency condition. Convergence analysis shows that the method yields the approximate solutions with optimal accuracy in L2(Ω) norm for u and in H( div;Ω) norm for the flux σ. Numerical experiments are given to verify the theoretical results.