Mixed and hybrid Petrov-Galerkin finite element discretization for optimal control of the wave equation
The article deals with a classical problem in optimal control of partial differential equation. The dynamics is given by the wave equation with Dirichlet boundary condition, with a distributed control acting on the whole domain. After rewriting the wave equation as a first-order coupled PDE with pressure and velocity variables, the cost to be minimized is a balance between the tracking of pressure-velocity towards a reference and the \(L^2\) cost of the control. The novelty of the article is to propose a numerical method that is based on the Raviart-Thomas finite elements in space and a Petrov-Galerkin scheme. The main advantage of these choices is that the discretize-then-optimize and optimize-then-discretize procedures coincide. Convergence of the scheme is proved to be linear.
scientific article; zbMATH DE number 7493702
scientific article