  A previously developed quantum reduced-order model is revised and applied,
together with the domain decomposition, to develop the quantum element method
(QEM), a methodology for fast and accurate simulation of quantum eigenvalue
problems. The concept of the QEM is to partition the simulation domain of a
quantum eigenvalue problem into smaller subdomains that are referred to as
elements. These elements could be the building blocks for quantum structures of
interest. Each of the elements is projected onto a functional space represented
by a reduced order model, which leads to a quantum Hamiltonian equation in the
functional space for each element. The basis functions in this study is
generated from proper orthogonal decomposition (POD). To construct a POD model
for a large domain, these projected elements are combined together, and the
interior penalty discontinuous Galerkin method is applied to stabilize the
numerical solution and to achieve the interface continuity. The POD is able to
optimize the basis functions (or POD modes) 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
Schr\"odinger equation. The proposed multi-element POD model (or QEM) is
demonstrated in several quantum-well structures with a focus on understanding
how to achieve accurate prediction of WFs with a small numerical DoF. It has
been shown that the QEM is able to achieve a substantial reduction in the DoF
with a high accuracy beyond the conditions accounted for in the training of the
POD modes.
