This paper develops a framework to include Dirichlet boundary conditions on a subset ofthe boundary which depends on time. In this model, the boundary conditions are weaklyenforced with the help of a Lagrange multiplier method. In order to avoid that the ansatzspace of the Lagrange multiplier depends on time, a bi-Lipschitz transformation, whichmaps a fixed interval onto the Dirichlet boundary, is introduced. An inf-sup condition aswell as existence results are presented for a class of second order initial-boundary valueproblems. For the semi-discretization in space, a finite element scheme is presented whichsatisfies a discrete stability condition. Because of the saddle point structure of theunderlying PDE, the resulting system is a DAE of index 3.
