Nitsche’s method for parabolic partial differential equations with mixed time varying boundary conditions

We investigate a finite element approximation of an initial boundary value problem associated with parabolic Partial Differential Equations endowed with mixed time varying boundary conditions, switching from essential to natural and viceversa. The switching occurs both in time and in different portions of the boundary. For this problem, we apply and extend the Nitsche’s method presented in [Juntunen and Stenberg, Mathematics of Computation, 2009] to the case of mixed time varying boundary conditions. After proving existence and numerical stability of the full discrete numerical solution obtained by using the θ-method for time discretization, we present and discuss a numerical test that compares our method to a standard approach based on remeshing and projection procedures.