A posteriori error estimation for finite element approximations of a PDE-constrained optimization problem in fluid dynamics
Provides rigorous error estimation for optimization problems in fluid dynamics, benefiting researchers in computational PDE-constrained optimization.
The paper derives globally reliable and locally efficient a posteriori error estimators for PDE-constrained optimization problems in fluid dynamics with control constraints, applicable to a wide range of finite element methods. Numerical examples illustrate the theory.
We derive globally reliable a posteriori error estimators for a PDE-constrained optimization problem involving linear models in fluid dynamics as state equation; control constraints are also considered. The corresponding local error indicators are locally efficient. The assumptions under which we perform the analysis are such that they can be satisfied for a wide variety of stabilized finite element methods as well as for standard finite element methods. When stabilized methods are considered, no a priori relation between the stabilization terms for the state and adjoint equations is required. If a lower bound for the inf-sup constant is available, a posteriori error estimators that are fully computable and provide guaranteed upper bounds on the norm of the error can be obtained. We illustrate the theory with numerical examples.