Nitsche's method  is a finite element technique, related to discontinuous  Galerkin methods which is predates,  that uses a weak formulation without Lagrange multipliers to take into account Dirichlet or contact boundary conditions. Although as such it often becomes non-conforming, and thus its well-posedness is not obvious, the method can also be derived from stabilised FEM for which the analysis is straightforward.  

In this talk, we will give an overview  on,  and show numerical results of, Nitsche's/stabilized FEM for contact problems.

Problems are first written in a mixed form, with the contact pressure acting as a Lagrange multiplier, and the stabilised formulation is derived by adding appropriately weighted residual terms to the discrete variational forms.

Given that the discrete formulation is uniformly stable, it leads to a quasi-optimal a priori error estimate without further regularity assumptions. Moreover,    local lower bounds ensure optimality of the a posteriori estimates.

Before implementation, the discrete Lagrange multiplier can be locally eliminated, which gives rise to Nitsche's method. 

References

[1] T. Gustafsson, P. Raback, and J. Videman. Mortaring for linear elasticity using mixed and stabilized finite elements. Comput. Methods Appl. Mech. Engrg. 404 (2023).

[2] T. Gustafsson, R. Stenberg, and J. Videman. Mixed and stabilized finite element methods for the obstacle problem. SIAM J.  Numer. Anal. 55 (2017).

[3] T. Gustafsson, R. Stenberg, and J. Videman. Error analysis of Nitsche's mortar method. Numer. Math.  142 (2019). 

[4] T. Gustafsson, R. Stenberg, and J. Videman. On Nitsche's method for elastic contact problems. SIAM J. Sci. Comput. 42 (2020).

[5] T. Gustafsson,  and J. Videman. Stabilized Finite Elements for Tresca friction problem. ESAIM: M2AN 56 (2022).

[6] T. Gustafsson,  and J. Videman. Stabilized Finite Element Method for Stokes problem with nonlinear slip condition.  BIT Numer. Math.  64 (2024).