| Abstract: |
| A symmetric DWDG method is used to discretize a control constrained elliptic optimal control problem with the PDE constraint as the Poisson`s equation. We develop a scheme to obtain a finite dimensional optimization problem which is then solved with a primal-dual active set strategy. The convergence of the numerical solution pair $(\overline{y_h},\overline{u_h})$ is proved. The rates of convergence are established in $L_2$ and energy norms. \
Next, a fully discrete scheme to solve the parabolic variational inequality with a general obstacle function in $\mathbb{R}^2$ that uses a symmetric dual-wind discontinuous Galerkin discretization in space and a backward Euler discretization in time is proposed and analyzed. The convergence of numerical solutions in $L^\infty(L^2)$ and $L^2(H^1)$ like energy norms is established and the rates are computed. Several numerical tests are provided to demonstrate the robustness and effectiveness of the proposed methods. This is a joint work with Tom Lewis, Aaron Rapp and Yi Zhang. |
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