Quantum element method for quantum eigenvalue problems derived from projection-based model order reduction

An effective multi-element simulation methodology for quantum eigenvalue problems is investigated. The approach is derived from a reduced-order model based on a data-driven learning algorithm, together with the concept of domain decomposition. The approach partitions the simulation domain of a quantum eigenvalue problem into smaller subdomains that, referred to as elements, could be the building blocks for quantum structures of interest. In this quantum element method (QEM), each element is projected onto a functional space represented by a set of basis functions (or modes) that are generated from proper orthogonal decomposition (POD). To construct a POD model for a large domain, these projected elements can be combined together, and the interior penalty discontinuous Galerkin method is applied to achieve the interface continuity and stabilize the numerical solution. The POD is able to optimize the basis functions specifically tailored to the geometry and parametric variations of the problem and can therefore substantially reduce the degree of freedom (DoF) needed to solve the Schrödinger equation. To understand the fundamental issues of the QEM, demonstrations in this study focus on examining the accuracy and DoF of the QEM influenced by the training settings for generation of POD modes, selection of the penalty number, suppression of interface discontinuities, structure size and complexity, etc. It has been shown that the QEM is able to achieve a substantial reduction in the DoF with a high accuracy even beyond the training conditions for the POD modes if the penalty number is selected within an appropriate range.