A High-Order Finite Element Framework for Solving Nonlinear Schrodinger Equations in Complex Domains

This paper introduces a new type of high order finite element method for numerically solving NLSEs in complex domains. NLSEs play a central role in the modelling of wave phenomena in optics, quantum mechanics, and fluid dynamics, however, finding their solution in geometrically complex domains is still a challenge due to nonlinearities and boundary complexities. The framework that is proposed is basing on high-order finite element discretization on unstructured meshes, which makes it possible to appropriately and flexibly compute with complex geometries. Key aspects are weak formulation with adaptation time integration, strong treatment of terms with nonlinearity by Newton-type iterations, and assignment of various other boundary conditions. Numerical experiments show high order convergence rates, conservation properties and better performance than low order methods. In order to validate the efficiency and accuracy of the framework, the soliton propagation, vortex dynamics and domains with irregular shapes are used as the benchmarks. This work leads to the development of numerical methods for NLSEs uncombining geometric flexibility and high accuracy which may provide a means for simulation in photonic devices, Bose-Einstein condensates, and other applications.