Adaptive finite element methods for linear-quadratic convection dominated elliptic optimal control problems
Master of Arts thesis
The numerical solution of linear-quadratic elliptic optimal control problems requires the solution of a coupled system of elliptic partial differential equations (PDEs), consisting of the so-called state PDE, the adjoint PDE and an algebraic equation. Adaptive finite element methods (AFEMs) attempt to locally refine a base mesh in such a way that the solution error is minimized for a given discretization size. This is particularly important for the solution of convection dominated problems where inner and boundary layers in the solutions to the PDEs need to be sufficiently resolved to ensure that the solution of the discretized optimal control problem is a good approximation of the true solution. This thesis reviews several AFEMs based on energy norm based error estimates for single convection dominated PDEs and extends them to the solution of the coupled system of convection dominated PDEs arising from the optimality conditions for optimal control problems. Keywords Adaptive finite element methods, optimal control problems, convection-diffusion equations, local refinement, error estimation.