A posteriori error estimates for discontinuous Galerkin approximation of non-stationary convection-diffusion optimal control problems

Abstract

In this paper, we investigate a discontinuous Galerkin finite element approximation of non-stationary convection dominated diffusion optimal control problems with control constraints. The state variable is approximated by piecewise linear polynomial space and the control variable is discretized by variational discretization concept. Backward Euler method is used for time discretization. With the help of elliptic reconstruction technique residual type a posteriori error estimates are derived for state variable and adjoint state variable, which can be used to guide the mesh refinement in the adaptive algorithm. Numerical experiment is presented, which indicates the good behaviour of the a posteriori error estimators.

The Optimal Dividend Payout Model with Terminal Values and Its Application

“…For this kind of problem, sometimes we cannot derive the explicit solutions of optimal value function. Then we need to use some suitable numerical methods as in Zhou et al [35] and Zhou [36] to simulate.…”

A Mixed Discontinuous Galerkin Approximation of Time Dependent Convection Diffusion Optimal Control Problem

“…Theorem 8. Let ( , q, , p) and ( ℎ ( ), q ℎ ( ), ℎ ( ), p ℎ ( )) be the solutions of (12) and (14), respectively. Then we have…”

“…Multiplying (114) by 2 and summing up with respect to from 1 to yield ℎ ( )) be the solutions of (12) and (92), respectively. Then we have…”

“…Then the theorem result follows from (122) and (124). ( , q, , p, ) and ( ℎ , q ℎ , −1 ℎ , p −1 ℎ , ℎ ) be the solutions of (12) and (92), respectively. Let…”

“…In [3][4][5][6] the stabilization method such as local projection stabilization, SUPG, and continuous interior penalty method are investigated. The discontinuous Galerkin approximation including primary discontinuous Galerkin method and local discontinuous Galerkin method is addressed in [7][8][9][10][11][12]. In [13] the authors discuss the characteristic finite element approximation of transient convection diffusion optimal control problem.…”