The spectral element (SE) and local Galerkin (LG) methods may be regarded as variants and generalizations of the classic Galerkin approach. In this study, the second-order spectral element (SE2) method is compared with the alternative LG scheme referred to as o2o3, which combines a second-order field representation (o2) with a third-order representation of the flux (o3). The full name of o2o3 is o2o3C⁰C¹, where the continuous basis functions in C⁰-space are used for the field representation and the piecewise third-order differentiable basis functions in C¹-space are used for the flux approximation. The flux in o2o3 is approximated by a piecewise polynomial function that is both continuous and differentiable, contrary to several Galerkin and LG schemes that use either continuous or discontinuous basis functions for flux approximations. We show that o2o3 not only has some advantages of SE schemes but also possesses third-order accuracy similar to o3o3 and SE3, while SE2 possesses second-order accuracy and does not show superconvergence. SE3 has an approximation order greater than or equal to three and uses the irregular Gauss-Lobatto collocation grid, whereas SE2 and o2o3 have a regular collocation grid; this constitutes an advantage for physical parameterizations and follow-up models, such as chemistry or solid-earth models. Furthermore, o2o3 has the technical simplicity of SE2. The common features (accuracy, convergence, and numerical dispersion relations) and differences between these schemes are described in detail for one-dimensional homogeneous advection tests. A two-dimensional test for cut cells indicates the suitability of o2o3 for realistic applications.