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A three-dimensional (3-D) spectral-element method (SEM) based on Gauss-Lobatto-Legendre (GLL) polynomials is proposed to solve the Schrödinger equation in nanodevice simulation. Galerkin's method is employed to obtain the system equation. The high-order basis functions employed are orthogonal and the numerical quadrature points are the same as the GLL integration points, leading to a diagonal mass matrix and a more sparse stiffness matrix. Thus, the proposed method leads to a regular eigenvalue problem, rather than a generalized eigenvalue problem, greatly reducing the computer-memory requirement and central-processing-unit (CPU) time in comparison with the conventional finite-element method (FEM). Furthermore, the SEM is implemented for high geometrical orders, where curved structures can be modeled up to the accuracy comparable to the interpolation accuracy afforded by the basis functions. Numerical examples verify a spectral accuracy with the interpolation orders, and confirm that higher geometrical orders are essential for curved structures to achieve overall spectral accuracy. Examples of quantum dots in various structures, including a waveguide, are analyzed with mixed boundary conditions. Numerical results show that the SEM is an efficient alternative to conventional FEM and to the finite-difference method (FDM) for nanodevice simulation.