J. Korean Math. Soc. 50 (2013), No. 1, pp. 81–93 http://dx.doi.org/10.4134/JKMS.2013.50.1.081 A CELL BOUNDARY ELEMENT METHOD FOR A FLUX CONTROL PROBLEM Youngmok Jeon and Hyung-Chun Lee Abstract. We consider a distributed optimal ﬂux control problem: ﬁnd- ing the potential of which gradient approximates the target vector ﬁeld under an elliptic constraint. Introducing the Lagrange multiplier and a change of variables the Euler-Lagrange equation turns into a coupled equation of an elliptic equation and a reaction diﬀusion equation. The change of variables reduces iteration steps dramatically when the Gauss- Seidel iteration is considered as a solution method. For the elliptic equa- tion solver we consider the Cell Boundary Element (CBE) method, which is the ﬁnite element type ﬂux preserving methods. 1. Introduction In this article, we consider a distributed optimal control problem for a second order partial diﬀerential equation: seek (u, p) ∈ H 1 (Ω)×L 2 (Ω) which minimizes the cost functional (1.1) J (u, p)= J u d (u)+ δN (p) subject to (1.2) −∇ · (a∇u)= p on Ω, where J u d (u) and N (p) are some (semi)norms of u and p, respectively and the parameter, δ is a positive regularization parameter. We assume that Ω = ∪ J j=1 Ω j is a simply connected polygonal domain with the boundary Γ. The permeability coeﬃcient, a is a positive deﬁnite, symmetric tensor and it is constant on each subdomain Ω j . In our problem the target function u d ∈ H 1 (Ω) will be used to provide the target ﬂux ﬁeld ∇u d and the Dirichlet boundary condition u = u d on Γ. Throughout this article we assume u = u d = 0 on Γ to simplify our analysis. Received November 22, 2011; Revised April 9, 2012. 2010 Mathematics Subject Classiﬁcation. 65M55, 65N30. Key words and phrases. cell boundary element method, optimal control problem, Gauss- Seidel iteration. This work was supported by Basic Science Research Program through the National Re- search Foundation of Korea(NRF) (grant number NRF 2010-0026032). c 2013 The Korean Mathematical Society 81