Mathematics

We introduce and discuss the Nitsche finite element method for the simplest elliptic problem: the Poisson problem with Neumman and Dirichlet mixed boundary conditions, using an extended mesh, and cut boundary elements. We focus on the a posteriori error estimates with emphasis in the cases of a quantity of interest a linear functional in the interior and a functional related to the normal flux in the Dirichlet boundary. The a posteriori error representation Lemma's and the corresponding error optimal bounds are presented, and the related sketch proof is demonstrated. We examine the case of uncertainties in our problem for a quantity of interest and an application related to the failure probability of a PDE model with random data with some ways to reduce the computational cost.