Finite Element Methods for Elastic Contact: Penalty and Nitsche

TL;DR

This paper analyzes two FEM strategies for elastic contact: the penalty method and Nitsche's method. The penalty approach imposes contact via a penalty term $\int_Γ \frac{1}{2\varepsilon}(u_1-u_2)^2 ds$, but modelling error prevents achieving the optimal convergence rate; Nitsche's method, obtained as a simple consistency correction and symmetrization of the penalty form, yields a stable, symmetric bilinear form and optimal convergence under a proper edgewise penalty scaling. The authors derive a priori error estimates $\|u-u_h\|_E \lesssim h^{s-1}\|u\|_s$ for $1\le s\le p+1$ and provide an a posteriori estimator, demonstrating reliability and efficiency, with numerical verifications on smooth and singular problems confirming the superior performance of Nitsche's method over the penalty approach. The work highlights the practical impact of Nitsche's method, including its simplicity, robustness, and adoption in major FEM codes, making it a preferred tool for elastic contact problems.

Abstract

We consider two methods for treating elastic contact problems with the finite element method; the penalty method and Nitsche's method. For the penalty method we discuss how the penalty parameter should be chosen. Both the theoretical analysis and numerical examples show that an optimal convergence rate cannot be achieved. The method is contrasted to that of Nitsche which is optimally convergent. We also give the derivation of Nitsche's method by a very simple consistency correction of the penalty method.

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