A PosterioriError Estimates of Semidiscrete Mixed Finite Element Methods for Parabolic Optimal Control Problems

Abstract

A posteriori error estimates of semidiscrete mixed finite element methods for quadratic optimal control problems involving linear parabolic equations are developed. The state and co-state are discretised by Raviart-Thomas mixed finite element spaces of order k, and the control is approximated by piecewise polynomials of order k (k ≥ 0). We derive our a posteriori error estimates for the state and the control approximations via a mixed elliptic reconstruction method. These estimates seem to be unavailable elsewhere in the literature, although they represent an important step towards developing reliable adaptive mixed finite element approximation schemes for the control problem.

Adaptive variational discretization approximation method for parabolic optimal control problems

“…In [24], the authors constructed an adaptive space-time FEM for POCPs. There are also some results on residual-type a posteriori error estimates of FEM or mixed FEM for POCPs which can be found in [6,23,31], where the authors do not give any adaptive FEM approximation for POCPs.…”

A priori and a posteriori error estimates of H1‐Galerkin mixed finite element method for parabolic optimal control problems

“…Thus, when our primary concern is to approximate both state and gradient variables, the mixed methods are preferable as they provide approximations to both the scalar and the flux variables. We refer to some studies for an introduction to the classical MFEM for optimal control problem and a review of related literature. In the classical MFEM, the finite element spaces used for the scalar and flux variables can not be chosen independently, and they must satisfy the Ladyzhenskaya‐Babuska‐Brezzi condition, and this restricts the choice of finite element spaces.…”