Superconvergence analysis of fully discrete finite element methods for semilinear parabolic optimal control problems. (English) Zbl 1330.49030
Summary: We study the superconvergence property of fully discrete finite element approximation for quadratic optimal control problems governed by semilinear parabolic equations with control constraints. The time discretization is based on difference methods, whereas the space discretization is done using finite element methods. The state and the adjoint state are approximated by piecewise linear functions and the control is approximated by piecewise constant functions. First, we define a fully discrete finite element approximation scheme for the semilinear parabolic control problem. Second, we derive the superconvergence properties for the control, the state and the adjoint state. Finally, we do some numerical experiments for illustrating our theoretical results.
MSC:
| 49M25 | Discrete approximations in optimal control |
| 35K58 | Semilinear parabolic equations |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
Keywords:
superconvergence property; quadratic optimal control problem; fully discrete finite element approximation; semilinear parabolic equation; interpolate operatorReferences:
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