Abstract
In this paper, we consider the mixed finite element methods for quadratic optimal control problems governed by convective diffusion equations. The state and the co-state are discretized by the lowest order Raviart-Thomas mixed finite element spaces and the control is approximated by piecewise constant functions. Using some proper duality problems, we derive a posteriori $L^{2}(0,T;L^{2}(\Omega))$ error estimates for the scalar functions. Such estimates, which are apparently not available in the literature, are an important step toward developing reliable adaptive mixed finite element approximation schemes for the control problem.
Highlights
As far as we know, optimal control problems [ ] have been extensively utilized in many aspects of the modern life such as social, economic, scientific, and engineering numerical simulation
A systematic introduction to finite element methods for PDEs and optimal control can be found for example in [ – ]
An adaptive finite element approximation ensures a higher density of nodes in a certain area of the given domain, where the solution is more difficult to approximate, indicated by a posteriori error estimators
Summary
As far as we know, optimal control problems [ ] have been extensively utilized in many aspects of the modern life such as social, economic, scientific, and engineering numerical simulation. They must be solved successfully with efficient numerical methods. The adaptive finite element method has been investigated extensively It has become one of the most popular methods in the scientific computation and numerical modeling. Mixed finite element methods are appropriate for the state equations in such cases since both the scalar variable and its flux variable can be approximated to the same accuracy by using such methods; see, for example, [ – ].
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
