Optimal Control of Stationary Doubly Diffusive Flows on Two and Three Dimensional Bounded Lipschitz Domains: Numerical Analysis
CoRR(2024)
摘要
In this work, we propose fully nonconforming, locally exactly divergence-free
discretizations based on lowest order Crouziex-Raviart finite element and
piecewise constant spaces to study the optimal control of stationary double
diffusion model presented in [Bürger, Méndez, Ruiz-Baier, SINUM (2019),
57:1318-1343]. The well-posedness of the discrete uncontrolled state and
adjoint equations are discussed using discrete lifting and fixed point
arguments, and convergence results are derived rigorously under minimal
regularity. Building upon our recent work [Tushar, Khan, Mohan arXiv (2023)],
we prove the local optimality of a reference control using second-order
sufficient optimality condition for the control problem, and use it along with
an optimize-then-discretize approach to prove optimal order a priori error
estimates for the control, state and adjoint variables upto the regularity of
the solution. The optimal control is computed using a primal-dual active set
strategy as a semi-smooth Newton method and computational tests validate the
predicted error decay rates and illustrate the proposed scheme's applicability
to optimal control of thermohaline circulation problems.
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