Worked Out Problem 7: A Nonlinear Elliptic Problem

In this chapter, we consider a scalar nonlinear elliptic PDE as a benchmark for complex material problems in a two-dimensional domain. The aim is to obtain efficient evaluations of the unknown field and the output of interest, defined as its average over the whole domain. The solution is approximated using reduced order modeling techniques based on the POD-Galerkin method and the Empirical Interpolation Method. The problem is described by presenting the physical domain, the goal, the parametrization, the boundary conditions, the governing PDE, and the corresponding weak formulation with its affine decomposition. Numerical results show the performance of the reduced approach with respect to the high-fidelity discretization, the quantity of interest, and the speedup. We provide the links to the source code, to the Jupyter notebook, and to the web-server ARGOS.