3. Distributed and boundary control problems for the semidiscrete Cahn–Hilliard/Navier–Stokes system with nonsmooth Ginzburg–Landau energies
-
M. Hintermüller
Abstract
This chapter is concerned with optimal control problems for the coupled Cahn-Hilliard (CH)/Navier-Stokes (NS) system related to ‘model H’ of Hohenberg and Halperin [20]. It proposes a time discretization allowing suitable energy estimates, that in particular force the total energy to decrease without control action, and it considers distributed aswell as boundary control of the fluid. For nonsmooth potentials, including the double-obstacle potential contained in the associated Ginzburg- Landau energy, a regularization procedure based on a mollified Moreau-Yosida approximation is applied. The resulting regularized problems are discretized by finite elements and solved via a gradient descent method. Several numerical examples document the behavior of the algorithm as well as the controlled CH-NS system for boundary and for distributed control.
Abstract
This chapter is concerned with optimal control problems for the coupled Cahn-Hilliard (CH)/Navier-Stokes (NS) system related to ‘model H’ of Hohenberg and Halperin [20]. It proposes a time discretization allowing suitable energy estimates, that in particular force the total energy to decrease without control action, and it considers distributed aswell as boundary control of the fluid. For nonsmooth potentials, including the double-obstacle potential contained in the associated Ginzburg- Landau energy, a regularization procedure based on a mollified Moreau-Yosida approximation is applied. The resulting regularized problems are discretized by finite elements and solved via a gradient descent method. Several numerical examples document the behavior of the algorithm as well as the controlled CH-NS system for boundary and for distributed control.
Kapitel in diesem Buch
- Frontmatter I
- Contents V
-
Part I
- 1. Geometric issues in PDE problems related to the infinity Laplace operator 3
- 2. Solution of free boundary problems in the presence of geometric uncertainties 20
- 3. Distributed and boundary control problems for the semidiscrete Cahn–Hilliard/Navier–Stokes system with nonsmooth Ginzburg–Landau energies 40
- 4. High-order topological expansions for Helmholtz problems in 2D 64
- 5. On a new phase field model for the approximation of interfacial energies of multiphase systems 123
- 6. Optimization of eigenvalues and eigenmodes by using the adjoint method 142
- 7. Discrete varifolds and surface approximation 159
-
Part II
- Preface 173
- 8. Weak Monge–Ampère solutions of the semi-discrete optimal transportation problem 175
- 9. Optimal transportation theory with repulsive costs 204
- 10. Wardrop equilibria: long-term variant, degenerate anisotropic PDEs and numerical approximations 257
- 11. On the Lagrangian branched transport model and the equivalence with its Eulerian formulation 281
- 12. On some nonlinear evolution systems which are perturbations of Wasserstein gradient flows 304
- 13. Pressureless Euler equations with maximal density constraint: a time-splitting scheme 333
- 14. Convergence of a fully discrete variational scheme for a thin-film equation 356
- 15. Interpretation of finite volume discretization schemes for the Fokker–Planck equation as gradient flows for the discrete Wasserstein distance 400
- Index 417
Kapitel in diesem Buch
- Frontmatter I
- Contents V
-
Part I
- 1. Geometric issues in PDE problems related to the infinity Laplace operator 3
- 2. Solution of free boundary problems in the presence of geometric uncertainties 20
- 3. Distributed and boundary control problems for the semidiscrete Cahn–Hilliard/Navier–Stokes system with nonsmooth Ginzburg–Landau energies 40
- 4. High-order topological expansions for Helmholtz problems in 2D 64
- 5. On a new phase field model for the approximation of interfacial energies of multiphase systems 123
- 6. Optimization of eigenvalues and eigenmodes by using the adjoint method 142
- 7. Discrete varifolds and surface approximation 159
-
Part II
- Preface 173
- 8. Weak Monge–Ampère solutions of the semi-discrete optimal transportation problem 175
- 9. Optimal transportation theory with repulsive costs 204
- 10. Wardrop equilibria: long-term variant, degenerate anisotropic PDEs and numerical approximations 257
- 11. On the Lagrangian branched transport model and the equivalence with its Eulerian formulation 281
- 12. On some nonlinear evolution systems which are perturbations of Wasserstein gradient flows 304
- 13. Pressureless Euler equations with maximal density constraint: a time-splitting scheme 333
- 14. Convergence of a fully discrete variational scheme for a thin-film equation 356
- 15. Interpretation of finite volume discretization schemes for the Fokker–Planck equation as gradient flows for the discrete Wasserstein distance 400
- Index 417
