A numerical method for handling boundary and transmission conditions

In this paper, we propose a numerical method for handling boundary or transmission conditions in some linear partial differential equations. Depending on the nature of the conditions-essential, like the Dirichlet condition, or natural, like the Neumann condition-we derive a formulation based on a Nitsche approach together with an original exchange approach. We present our method first in a model problem, the Laplace problem, for which the Nitsche method was introduced to impose weakly essential boundary conditions. Then, we illustrate the method with two examples. First, the Maxwell equations, where the method shows its ability to handle singular solutions in non-convex domains. Then, an elasticity problem in a layered non-homogeneous domain, where the method proves to correctly take into account the transmission conditions at the interface between the layers and in the presence of cracks.