Robust preconditioning and error estimates for optimal control of the convection-diffusion-reaction equation with limited observation in Isogeometric analysis

TL;DR

This paper develops robust preconditioning techniques and error estimates for an optimal control problem governed by a convection-diffusion-reaction equation, utilizing Isogeometric Analysis for discretization and multigrid methods for efficiency.

Contribution

It introduces a new preconditioner robust to problem parameters and provides error analysis for a specific discretization scheme using Isogeometric Analysis.

Findings

Preconditioner is robust with respect to problem parameters.

A priori error estimates are established for the discretization.

Macro Gauss-Seidel smoother improves multigrid efficiency.

Abstract

In this paper we analyze an optimization problem with limited observation governed by a convection--diffusion--reaction equation. Motivated by a Schur complement approach, we arrive at continuous norms that enable analysis of well-posedness and subsequent derivation of error analysis and a preconditioner that is robust with respect to the parameters of the problem. We provide conditions for inf-sup stable discretizations and present one such discretization for box domains with constant convection. We also provide a priori error estimates for this discretization. The preconditioner requires a fourth order problem to be solved. For this reason, we use Isogeometric Analysis as a 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.