A CELL BOUNDARY ELEMENT METHOD FOR A FLUX CONTROL PROBLEM Jeon, Youngmok; Lee, Hyung-Chun;
We consider a distributed optimal flux control problem: finding 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 equation solver we consider the Cell Boundary Element (CBE) method, which is the finite element type flux preserving methods.
cell boundary element method;optimal control problem;Gauss-Seidel iteration;
S. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, Springer-Verlag, Berlin, New York, 1994.
A. Fursikov, Optimal Control of Distributed systems, Theory and Applications, American Mathematical Society, Providence, RI, 2000.
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, 1998.
M. Gunzburger, Perspectives in Flow Control and Optimization, SIAM, Philadelphia, 2003.
Max D. Gunzburger and H. C. Lee, Analysis, approximation, and computation of a coupled solid/fluid temperature control problem, Comput. Methods Appl. Mech. Engrg. 118 (1994), no. 1-2, 133-152.
Y. Jeon, E.-J. Park, Nonconforming cell boundary element methods for elliptic problems on triangular mesh, Appl. Numer. Math. 58 (2008), no. 6, 800-814.
Y. Jeon and D. Sheen, Analysis of a cell boundary element method, Adv. Comput. Math. 22 (2005), no. 3, 201-222.
J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications. Vol. I, Springer, New York, 1972.
A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations, Springer, 1994.