Numerical Approximation of Large Contrast Problems with the Unfitted Nitsche Method

These notes are concerned with the numerical treatment of the coupling between second order elliptic problems that feature large contrast between their characteristic coefficients. In particular, we study the application of Nitsche’s method to set up a robust approximation of interface conditions in the framework of the finite element method. The notes are subdivided in three parts. Firstly, we review the weak enforcement of Dirichlet boundary conditions with particular attention to Nitsche’s method and we discuss the extension of such technique to the coupling of Poisson equations. Secondly, we review the application of Nitsche’s method to large contrast problems, discretised on computational meshes that capture the interface of discontinuity between coefficients. Finally, we extend the previous schemes to the case of unfitted meshes, which occurs when the computational mesh does not conform with the interface between subproblems.

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