Superconvergence of triangular mixed finite elements for optimal control problems with an integral constraint

Zhou, Jianwei; Chen, Yanping; Dai, Yongquan

doi:10.1016/j.amc.2010.07.006

Cited by 17 publications

(12 citation statements)

References 19 publications

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“…[6][7][8][9][10][11], for example. However, it is well known that the choice of mixed element spaces for the state variable and its flux must strictly satisfy LBB consistency condition, and a typical experience is to use the RaviartThomas (RT) elements [12].…”

Section: Introductionmentioning

confidence: 99%

A Stabilized Mixed Finite Element Method for Elliptic Optimal Control Problems

Rui

Hou

et al. 2015

J Sci Comput

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In this paper, we propose a new mixed finite element method, called stabilized mixed finite element method, for the approximation of optimal control problems constrained by a first-order elliptic system. This method is obtained by adding suitable elementwise least-squares residual terms for the primal state variable y and its flux σ . We prove the coercive and continuous properties for the new mixed bilinear formulation at both continuous and discrete levels. Therefore, the finite element function spaces do not require to satisfy the Ladyzhenkaya-Babuska-Brezzi consistency condition. Furthermore, the state and flux state variables can be approximated by the standard Lagrange finite element. We derive optimality conditions for such optimal control problems under the concept of Discretizationthen-Optimization, and then a priori error estimates in a weighted norm are discussed. Finally, numerical experiments are given to confirm the efficiency and reliability of the stabilized method.

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Section: Introductionmentioning

confidence: 99%

A Stabilized Mixed Finite Element Method for Elliptic Optimal Control Problems

Rui

Hou

et al. 2015

J Sci Comput

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“…where 1 , 2 denote the triangular Δ1 ′ 2 ′ 3 ′ (where nodes 1 and 1 ′ overlap), the quadrilateral □2 ′ 233 ′ , respectively, and f 1 i , f 2 i are quadrature functions in 1 , 2 , respectively, and i , i = 1, 2, 3 is the basis function for node i. Then some proper numerical quadratures are used in (23). So do all examples in this section.…”

Section: No Scalingmentioning

confidence: 99%

“…There have been extensive studies in numerical methods for diffusion problems, such as the finite difference method [3,6,7], the finite element method [8], and the finite volume element method [8][9][10][11]. The mixed finite element method (MFEM) has gained great popularity in the last several decades for the reason that they provide very accurate approximations of the primary unknown and its flux and they conserve mass locally [12][13][14][15][16][17][18][19][20][21][22][23]. In a mixed finite element formulation, both displacements and stresses (the derived 984 C. NIE AND H. YU flux) are approximated simultaneously.…”

Section: Introductionmentioning

confidence: 99%

A Raviart-Thomas mixed finite element scheme for the two-dimensional three-temperature heat conduction problems

Nie

Hai-yuan

2017

Int. J. Numer. Meth. Engng

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Summary For the two‐dimensional three‐temperature radiative heat conduction problem appearing in the inertial confinement numerical stimulations, we choose the Freezing coefficient method to linearize the nonlinear equations, and initially apply the well‐known mixed finite element scheme with the lowest order Raviart–Thomas element associated with the triangulation to the linearized equations, and obtain the convergence with one order with respect to the space direction for the temperature and flux function approximations, and design a simple but efficient algorithm for the discrete system. Three numerical examples are displayed. The former two verify theoretical results and show the super‐convergence for temperature and flux functions at the barycenter of the element, which is helpful for solving the radiative heat conduction problems. The third validates the robustness of this scheme with small energy conservative error and one order convergence for the time discretization. Copyright © 2016 John Wiley & Sons, Ltd.

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“…To obtain a high accuracy approximation of the derivative, mixed formulae and mixed finite element methods (MFEMs, for short) are introduced in lots of studies (see e.g. [3,16,24,30] and the references cited therein). The authors employed mixed finite element methods to investigate the 4th order equations in [12].…”

Section: Introductionmentioning

confidence: 99%