<p>In this paper, we investigate Petrov–Galerkin spectral element spectral method for mixed inhomogeneous boundary value problems defined on polyhedrons. Some results on three-dimensional Legendre quasi-orthogonal approximation in certain Jacobi weighted Sobolev spaces are established. These results play an important role in numerical solutions of partial differential equations defined on polyhedrons. As examples of applications, spectral element schemes are provided for two model problems, with the global spectral accuracy. Efficient numerical implementations are described. Numerical results demonstrate the high efficiency of suggested algorithms.</p>