## convdist1 details:

Keywords: analytic solution

Geometry: easy, fixed

Design: coupled via volume data

Differential operator:

• Convection-Diffusion:
• linear elliptic operator of order 2.
• Defined on a 2-dim domain in 2-dim space
• No time dependence.

Design constraints:

• none

State constraints:

• none

Mixed constraints:

• none

Submitted on 2013-02-06 by Roland Herzog. Published on 2013-02-12

## convdist1 description:

### Introduction

This problem is a prototypical example for a convection dominated stationary diﬀusion problem. Such problems are known to produce boundary layers and to lead to unphysical oscillatory solutions in conventional discretization schemes.

The present problem was originally proposed in [Collis and Heinkenschloss2002, Example 3] and discretized using an SUPG (streamline upwind Petrov Galerkin) approach. Discontinuous Galerkin schemes for the same problem were analyzed in Yücel et al. [2013].

### Variables & Notation

#### Given Data

The given data is chosen in a way which admits an analytic solution. The diﬀusion parameter $𝜀>0$ and the constant convection direction $c$ can be freely chosen. In Collis and Heinkenschloss [2002] and Yücel et al. [2013], the values $𝜀=1{0}^{-2}$ and

were used.

### Optimality System

The following optimality system for the state $y\in {H}_{0}^{1}\left(\Omega \right)$, the control $u\in {L}^{2}\left(\Omega \right)$, the adjoint state $p\in {H}_{0}^{1}\left(\Omega \right)$ characterizes the unique minimizer.

Note that the coeﬃcient $\left(divc\right)$ in the adjoint equation vanishes for the given constant velocity ﬁeld $c$.

### Supplementary Material

The optimal state, adjoint state, control and state constraint multiplier are known analytically:

 $\begin{array}{ccc}\hfill y& =\eta \left({x}_{1}\right)\phantom{\rule{0.3em}{0ex}}\eta \left({x}_{2}\right)\hfill & \hfill \\ \hfill p& =\xi \left({x}_{1}\right)\phantom{\rule{0.3em}{0ex}}\xi \left({x}_{2}\right)\hfill \\ \hfill u& =p.\hfill \end{array}$

Note that the boundary layer for the state lies near the right and upper boundary (where ${x}_{1}=1$ or ${x}_{2}=1$), while the boundary layer for the adjoint state is located on the opposite parts of the boundary.

### References

S. S. Collis and M. Heinkenschloss. Analysis of the streamline upwind/Petrov Galerkin method applied to the solution of optimal control problems. Technical Report CAAM TR02–01, Rice University, 2002. URL http://www.caam.rice.edu/heinken/papers/supg_analysis.pdf.

H. Yücel, M. Heinkenschloss, and B. Karasözen. Distributed optimal control of diﬀusion-convection-reaction equations using discontinuous Galerkin methods. Numerical Mathematics and Advanced Applications, pages 389–397, 2013. doi: 10.1007/978-3-642-33134-3_42.