On Nitsche's method for elastic contact problems

We show quasi-optimality and a posteriori error estimates for the frictionless contact problem between two elastic bodies with a zero-gap function. The analysis is based on interpreting Nitsche's method as a stabilised finite element method for which the error estimates can be obtained with minimal regularity assumptions and without the saturation assumption. We present three different Nitsche's mortaring techniques for the contact boundary each corresponding to a different stabilising term. Our numerical experiments show the robustness of Nitsche's method and corroborates the efficiency of the a posteriori error estimators

Nitsche's method for Kirchhoff plates

We introduce a Nitsche's method for the numerical approximation of the Kirchhoff-Love plate equation under general Robin-type boundary conditions. We analyze the method by presenting a priori and a posteriori error estimates in mesh-dependent norms. Several numerical examples are given to validate the approach and demonstrate its properties

The multi-level Monte Carlo finite element method for a stochastic Brinkman Problem

We present the formulation and the numerical analysis of the Brinkman problem derived in Allaire (Arch Rational Mech Anal 113(3): 209-259,1990. doi: 10.1007/BF00375065 , Arch Rational Mech Anal 113(3): 261-298, 1990. doi: 10.1007/BF00375066 ) with a lognormal random permeability. Specifically, the permeability is assumed to be a lognormal random field taking values in the symmetric matrices of size $$d\times d$$ , where $$d$$ denotes the spatial dimension of the physical domain $$D$$ . We prove that the solutions admit bounded moments of any finite order with respect to the random input's Gaussian measure. We present a Mixed Finite Element discretization in the physical domain $$D$$ , which is uniformly stable with respect to the realization of the lognormal permeability field. Based on the error analysis of this mixed finite element method (MFEM), we develop a multi-level Monte Carlo (MLMC) discretization of the stochastic Brinkman problem and prove that the MLMC-MFEM allows the estimation of the statistical mean field with the same asymptotical accuracy versus work as the MFEM for a single instance of the stochastic Brinkman problem. The robustness of the MFEM implies in particular that the present analysis also covers the Darcy diffusion limit. Numerical experiments confirm the theoretical result