A Splitting Least-squares Mixed Finite Element Method for Elliptic Optimal Control Problems

نویسندگان

  • HONGFEI FU
  • HONGXING RUI
  • HUI GUO
  • JIANSONG ZHANG
  • JIAN HOU
چکیده

In this paper, we propose a splitting least-squares mixed finite element method for the approximation of elliptic optimal control problem with the control constrained by pointwise inequality. By selecting a properly least-squares minimization functional, we derive equivalent two independent, symmetric and positive definite weak formulation for the primal state variable and its flux. Then, using the first order necessary and also sufficient optimality condition, we deduce another two corresponding adjoint state equations, which are both independent, symmetric and positive definite. Also, a variational inequality for the control variable is involved. For the discretization of the state and adjoint state equations, either RT mixed finite element or standard C0 finite element can be used, which is not necessary subject to the Ladyzhenkaya-Babuska-Brezzi condition. Optimal a priori error estimates in corresponding norms are derived for the control, the states and adjoint states, respectively. Finally, we use some numerical examples to validate the theoretical analysis.

برای دانلود رایگان متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

Analysis and approximation of optimal control problems for first-order elliptic systems in three dimensions

We examine analytical and numerical aspects of optimal control problems for firstorder elliptic systems in three dimensions. The particular setting we use is that of divcurl systems. After formulating some optimization problems, we prove the existence and uniqueness of the optimal solution. We then demonstrate the existence of Lagrange multipliers and derive an optimality system of partial diff...

متن کامل

Uniform Convergence of a Monotone Iterative Method for a Nonlinear Reaction-Diffusion Problem

The dynamics of matrix coupling with an application to Krylov methods p. 14 High precision method for calculating the energy values of the hydrogen atom in a strong magnetic field p. 25 Splitting methods and their application to the abstract Cauchy problems p. 35 Finite difference approximation of an elliptic interface problem with variable coefficients p. 46 The finite element method for the N...

متن کامل

Superconvergence of Least-squares Mixed Finite Elements

In this paper we consider superconvergence and supercloseness in the least-squares mixed finite element method for elliptic problems. The supercloseness is with respect to the standard and mixed finite element approximations of the same elliptic problem, and does not depend on the properties of the mesh. As an application, we will derive more precise a priori bounds for the least squares mixed ...

متن کامل

Analysis of First-Order System Least Squares (FOSLS) for Elliptic Problems with Discontinuous Coefficients: Part I

First-order system least squares (FOSLS) is a recently developed methodology for solving partial differential equations. Among its advantages are that the finite element spaces are not restricted by the inf-sup condition imposed, for example, on mixed methods and that the least-squares functional itself serves as an appropriate error measure. This paper studies the FOSLS approach for scalar sec...

متن کامل

Least-Squares Finite Element Methods for Optimality Systems Arising in Optimization and Control Problems

The approximate solution of optimization and optimal control problems for systems governed by linear, elliptic partial differential equations is considered. Such problems are most often solved using methods based on applying the Lagrange multiplier rule to obtain an optimality system consisting of the state system, an adjoint-state system, and optimality conditions. Galerkin methods applied to ...

متن کامل

ذخیره در منابع من


  با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید

عنوان ژورنال:

دوره   شماره 

صفحات  -

تاریخ انتشار 2016