Analysis for the space-time a posteriori error estimates for mixed finite element solutions of parabolic optimal control problems

• Published in: Numerical algorithms Vol. 96; no. 2; pp. 879 - 924
• Main Authors: Shakya, Pratibha, Kumar Sinha, Rajen
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.06.2024; Springer Nature B.V
• Subjects: Algebra; Algorithms; Approximation; Computer Science; Discretization; Error analysis; Estimates; Estimators; Finite element method; Implicit methods; Numeric Computing; Numerical Analysis; Optimal control; Original Paper; Spacetime; Theory of Computation; Variables; The backward-Euler method; 65N15; Mixed finite element method; Variational discretization; 49J20; A posteriori error estimates; Parabolic optimal control problems; 65N30
• ISSN: 1017-1398; 1572-9265
• DOI: 10.1007/s11075-023-01669-9

This paper investigates the space-time residual-based a posteriori error bounds of the mixed finite element method for the optimal control problem governed by the parabolic equation in a bounded convex domain. For the spatial discretization of the state and co-state variables, the lowest-order Raviart-Thomas spaces are utilized, although for the control variable, variational discretization technique is used. The backward-Euler implicit method is applied for temporal discretization. To provide a posteriori error estimates for the state and control variables in the L ∞ ( L 2 ) -norm, an elliptic reconstruction approach paired with an energy strategy is utilized. The reliability and efficiency of the a posteriori error estimators are discussed. The effectiveness of the estimators is finally confirmed through the numerical tests.