Robust Preconditioning: Optimal control of the convection-diffusion equation with limited observation

MSc Jarle Sogn

Nov. 24, 2020, 3:15 p.m. ZOOM

We consider an optimization problem with a Convection--Diffusion--Reaction equation as constraint. A Schur complement preconditioner is proposed and we show that the problem is well-posed. The preconditioner is robust with respect to all the problem parameters and the condition number of the preconditioned problem bounded by 4.05. We provide conditions for inf-sup stable discretizations and present one such discretization for box domains with constant convection.

The preconditioner requires a fourth order problem to be solved. For this reason, we use Isogeometric Analysis as method of discretization. To efficiently realize the preconditioner we consider geometric multigrid with a standard Gauss-Seidel smoother as well as a new macro Gauss-Seidel smoother. The latter smoother provides good results with respect to both the geometry mapping and the polynomial degree.