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Abstract: 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. [Copyright &y& Elsevier]

Abstract: This work deals with the crack problem simulation in dissimilar media. It proposes a new numerical method derived from a Nitsche approach for handling interface conditions in the elasticity equations. The Nitsche method, introduced to impose weakly essential boundary conditions in the scalar Laplace operator, has been then worked out more generally and transferred to continuity conditions. We propose here an extension of this method to the Navier–Lame equations. We derive a variational formulation that provides the solution in terms of displacements field in the case of a crack existence in a plate domain Ω, made of several different layers characterized by different material properties. We formulate the method for both the homogeneous and the dissimilar material domains and report some numerical experiments. [Copyright &y& Elsevier]

Abstract: In this paper, we propose a method derived from a Nitsche approach for handling boundary and transmission conditions in some partial differential equations. Several years ago, the Nitsche method was introduced to impose weakly essential boundary conditions in the scalar Laplace operator. We propose here an extension to vector div −curl problems. Two examples of applications are presented. The first one is concerned with the Maxwell equations. This allows us to solve these equations, particularly in domains with reentrant corners, where the solution can be singular. The second example deals with the Navier-Lame equations. One can handle the case of a crack existence in a plate domain made of several different layers, characterized by different material properties. Numerical experiments are reported. [Copyright &y& Elsevier]