Superconvergence of splitting positive definite mixed finite element for parabolic optimal control problems

In this paper, we investigate the superconvergence of fully discrete splitting positive definite mixed finite element (MFE) methods for parabolic optimal control problems. For the space discretization, the state and co-state are approximated by the lowest order Raviart–Thomas MFE spaces and the control variable is approximated by piecewise constant functions. The time discretization of the state and co-state are based on finite difference methods. We derive the superconvergence between the projections of exact solutions and numerical solutions or the exact solutions and postprocessing numerical solutions for the control, state and co-state. A numerical example is provided to validate the theoretical results.     

Superconvergence analysis of fully discrete finite element methods for semilinear parabolic optimal control problems

We study the superconvergence property of fully discrete finite element approximation for quadratic optimal control problems governed by semilinear parabolic equations with control constraints. The time discretization is based on difference methods, whereas the space discretization is done using finite element methods. The state and the adjoint state are approximated by piecewise linear functions and the control is approximated by piecewise constant functions. First, we define a fully discrete finite element approximation scheme for the semilinear parabolic control problem. Second, we derive the superconvergence properties for the control, the state and the adjoint state. Finally, we do some numerical experiments for illustrating our theoretical results.     

Superconvergence of mixed finite element methods for optimal control problems

In this paper, we investigate the superconvergence property of the numerical solution of a quadratic convex optimal control problem by using rectangular mixed finite element methods. The state and co-state variables are approximated by the lowest order Raviart-Thomas mixed finite element spaces and the control variable is approximated by piecewise constant functions. Some realistic regularity assumptions are presented and applied to error estimation by using an operator interpolation technique. We derive superconvergence properties for the flux functions along the Gauss lines and for the scalar functions at the Gauss points via mixed projections. Moreover, global superconvergence results are obtained by virtue of an interpolation postprocessing technique. Thus, based on these superconvergence estimates, some asymptotic exactness a posteriori error estimators are presented for the mixed finite element methods. Finally, some numerical examples are given to demonstrate the practical side of the theoretical results about superconvergence.

     

Superconvergence of a combined mixed finite element and discontinuous Galerkin approximation for an incompressible miscible displacement problem

A combined mixed finite element and discontinuous Galerkin approximation for an incompressible miscible displacement problem which includes molecular diffusion and dispersion in porous media is studied. That is to say, the mixed finite element method is applied to the flow equation, and the transport equation is solved by an interior penalty discontinuous Galerkin method. Convolution of the Darcy velocity approximation with the Bramble-Schatz kernel function and averaging are applied in the evaluation of the coefficients in the Galerkin procedure for the concentration. A superconvergence estimate is obtained. Numerical experimental results are presented to verify the theoretical analysis.     

Superconvergence for triangular finite elements

Based on two classes of the orthogonal expansions in a triangle, superconvergence of m-degree triangular finite element solution (for evenm) and its average gradient (for oddm) at symmetric points for a second order elliptic problem are studied. There are no other superconvergence points independent of the coefficients of elliptic equation. Project supported by the National Natural Science Foundation of China (Grant No. 19331021).     

Ciarlet–Raviart mixed finite element approximation for an optimal control problem governed by the first bi-harmonic equation

The Ciarlet–Raviart mixed finite element approximation is constructed to solve the constrained optimal control problem governed by the first bi-harmonic equation. The optimality conditions consisting of the state and the co-state equations is derived. Also, the a priori error estimates are analyzed. In the analysis of the a priori error estimates, the improved convergent rate of the higher order than existed results is proved. Some numerical experiments are performed to confirm the theoretical analysis for the a priori error estimate.     

A robust optimal preconditioner for the mixed finite element discretization of elliptic optimal control problems

In this paper, we consider the efficient solving of the resulting algebraic system for elliptic optimal control problems with mixed finite element discretization. We propose a block‐diagonal preconditioner for the symmetric and indefinite algebraic system solved with minimum residual method, which is proved to be robust and optimal with respect to both the mesh size and the regularization parameter. The block‐diagonal preconditioner is constructed based on an isomorphism between appropriately chosen solution space and its dual for a general control problem with both state and gradient state observations in the objective functional. Numerical experiments confirm the efficiency of our proposed preconditioner.     

A superconvergent $L^{\infty}$-error estimate of RT1 mixed methods for elliptic control problems with an integral constraint

In this paper, we investigate the superconvergence property of mixed finite element methods for a linear elliptic control problem with an integral constraint. The state and co-state are approximated by the order $k=1$ Raviart-Thomas mixed finite element spaces and the control variable is approximated by piecewise constant functions. A superconvergent approximation of the control variable $u$ will be constructed by a projection of the discrete adjoint state. It is proved that this approximation have convergence order $h^{2}$ in $L^{\infty}$-norm. Finally, a numerical example is given to demonstrate the theoretical results.     

Convergence and quasi-optimality of an adaptive finite element method for optimal control problems with integral control constraint

In this paper we study the convergence of an adaptive finite element method for optimal control problems with integral control constraint. For discretization, we use piecewise constant discretization for the control and continuous piecewise linear discretization for the state and the co-state. The contraction, between two consecutive loops, is proved. Additionally, we find the adaptive finite element method has the optimal convergence rate. In the end, we give some examples to support our theoretical analysis.     

Approximation of optimal control problems with state constraints

We study the approximation of control problems governed by elliptic partial differential equations with pointwise state constraints. For a finite dimensional approximation of the control set and for suitable perturbations of the state constraints, we prove that the corresponding sequence of discrete control problems converges to a relaxed problem. A similar analysis is carried out for problems in which the state equation is discretized by a finite element method.     

Superconvergence of tricubic block finite elements

In this paper, we first introduce interpolation operator of projection type in three dimen- sions, from which we derive weak estimates for tricubic block finite elements. Then using the estimate for the W 2, 1-seminorm of the discrete derivative Green’s function and the weak estimates, we show that the tricubic block finite element solution uh and the tricubic interpolant of projection type Πh3u have superclose gradient in the pointwise sense of the L∞-norm. Finally, this supercloseness is applied to superc...     

Superconvergence and a posteriori error estimates of RT1 mixed methods for elliptic control problems with an integral constraint

In this paper, we investigate the superconvergence property and a posteriori error estimates of mixed finite element methods for a linear elliptic control problem with an integral constraint. The state and co-state are approximated by the order k = 1 Raviart-Thomas mixed finite element spaces and the control variable is approximated by piecewise constant functions. Approximations of the optimal control of the continuous optimal control problem will be constructed by a projection of the discrete adjoint state. It is proved that these approximations have convergence order h ². Moreover, we derive a posteriori error estimates both for the control variable and the state variables. Finally, a numerical example is given to demonstrate the theoretical results.     

Finite element methods for elliptic optimal control problems with boundary observations

We study in this paper the finite element approximations to elliptic optimal control problems with boundary observations. The main feature of this kind of optimal control problems is that the observations or measurements are the outward normal derivatives of the state variable on the boundary, this reduces the regularity of solutions to the optimal control problems. We propose two kinds of finite element methods: the standard FEM and the mixed FEM, to efficiently approximate the underlying optimal control problems. For both cases we derive a priori error estimates for problems posed on polygonal domains. Some numerical experiments are carried out at the end of the paper to support our theoretical findings.     

A characteristic-mixed finite element method for time-dependent convection-diffusion optimal control problem

In this paper, we examine the method of characteristic-mixed finite element for the approximation of convex optimal control problem governed by time-dependent convection-diffusion equations with control constraints. For the discretization of the state equation, the characteristic finite element is used for the approximation of the material derivative term (i.e., the time derivative term plus the convection term), and the lowest-order Raviart-Thomas mixed element is applied for the approximation of the diffusion term. We derive some a priori error estimates for both the state and control approximations.     

Superconvergence of H 1-Galerkin mixed finite element methods for parabolic problems

In this article, we study the semidiscrete H ¹-Galerkin mixed finite element method for parabolic problems over rectangular partitions. The well-known optimal order error estimate in the L ²-norm for the flux is of order 𝒪(h ^k+1) (SIAM J. Numer. Anal. 35 (2), (1998), pp. 712–727), where k ≥ 1 is the order of the approximating polynomials employed in the Raviart–Thomas element. We derive a superconvergence estimate of order 𝒪(h ^k+3) between the H ¹-Galerkin mixed finite element approximation and an appropriately defined local projection of the flux variable when k ≥ 1. A the new approximate solution for the flux with superconvergence of order 𝒪(h ^k+3) is realized via a postprocessing technique using local projection methods.     

二阶椭圆方程Dirichlet边值问题混合元的超收敛

关于二阶椭圆方程Dirichlet边值问题混合元的超收敛,在正则矩形网格上,林群和林甲富在文[1]中,采用一阶Raviart-Thomas混合元空间,对有限元解经后处理后,其收敛于精确解的速度从二阶提高到四阶.本文拟将这一结果进行推广,讨论二阶椭圆方程Dirichlet边值问题的k阶Raviart-Thomas混合元的超收敛,得到了以k 3阶速度收敛于精确解的有限元解.     

Raviart-Thomas混合元的超收敛

考虑二阶椭圆方程Dirichlet边值问题在正则矩形网格上k阶RaviartThomas混合有限元的超收敛.对有限元解经插值处理后,与通常的有限元最优误差估计相比,收敛速度提高了两阶.