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A priori and a posteriori error estimates of H 1 ‐Galerkin mixed finite element method for parabolic optimal control problems
Author(s) -
Shakya Pratibha,
Sinha Rajen Kumar
Publication year - 2017
Publication title -
optimal control applications and methods
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.458
H-Index - 44
eISSN - 1099-1514
pISSN - 0143-2087
DOI - 10.1002/oca.2312
Subject(s) - mathematics , finite element method , a priori and a posteriori , piecewise , galerkin method , norm (philosophy) , mixed finite element method , state variable , backward euler method , discontinuous galerkin method , control variable , mathematical analysis , optimal control , mathematical optimization , euler equations , philosophy , physics , epistemology , political science , law , thermodynamics , statistics
Summary In this exposition, we study both a priori and a posteriori error analysis for the H 1 ‐Galerkin mixed finite element method for optimal control problems governed by linear parabolic equations. The state and costate variables are approximated by the lowest order Raviart‐Thomas finite element spaces, whereas the control variable is approximated by piecewise constant functions. Compared to the standard mixed finite element procedure, the present method is not subject to the Ladyzhenskaya‐Babuska‐Brezzi condition and the approximating finite element spaces are allowed to be of different degree polynomials. A priori error analysis for both the semidiscrete and the backward Euler fully discrete schemes are analyzed, and L ∞ ( L 2 ) convergence properties for the state variables and the control variable are obtained. In addition, L 2 ( L 2 )‐norm a posteriori error estimates for the state and control variables and L ∞ ( L 2 ) ‐norm for the flux variable are also derived.