Discontinuous Galerkin Methods and Model Reduction for the nonlinear Schrödinger Equation with FEniCS and DUNE

Abstract

We face the numerical solving process of the nonlinear Schrödinger equation (NLSE), also called Gross-Pitaevskii equation, which is a central semilinear partial differential equation (PDE) in the field of quantum mechanics. Concerning the spatial discretization, we compare the continuous Galerkin (CG) methods with their close relatives, the discontinuous Galerkin (DG) methods. As a practical example, we have a look at the symmetric interior penalty Galerkin (SIPG) method and the standard finite element method (FEM) with piecewise continuous basis functions. The semidiscrete NLSE is solved with the average vector field (AVF) method. A special focus is dedicated to the mass and energy preservation properties of the methods. Further on, we deal with model order reduction, carried out with proper orthogonal decomposition (POD) and randomized singular value decomposition (rSVD). The implementation is made in Python, using also the software FEniCS for CG and DUNE for DG methods.