Large eddy simulation (LES) is now over half a century old and while it has become more widely used as computational capabilities have expanded, its adoption as an engineering tool has arguably been limited by the shortcomings of subgrid models. Most current subgrid models are formulated under the assumption that the subgrid scales are approximately isotropic, and that other complications, such as numerical discretization and inhomogeneous resolution, are negligible. This limits the fidelity of the models when applied in complex flows. For LES to become a robust engineering tool, subgrid models applicable to more complex scenarios will be required. In particular, the effects of numerical discretization must be considered. In this thesis we develop several analytical and computational tools for identifying the characteristics of an LES introduced by numerical discretization and filtering. First, the effects of numerical dispersion error on the turbulent energy cascade are explored. It is shown that dispersion error due to convection by a large mean velocity causes a decoherence of the phase relationship among interacting Fourier modes, resulting in a reduction of the energy transfer rate from large to small scales. Nonlinear dispersion error due to convection from turbulent fluctuations is also explored through the development of an eddy-damped quasi-normal markovian (EDQNM) type of analysis that is applicable to the filtered turbulence in an LES. EDQNM is shown to be a useful tool for exploring dispersion effects because it exposes the relaxation rate of the third-order velocity correlations. An explicit filtering formulation based on the properties of the underlying numerics is developed to remove the highly dispersive wavemodes in an LES. Further, the EDQNM LES theory is also used to determine the a priori properties of the subgrid stress needed to recover an inertial range spectrum in the presence of non-spectral numerics and non-cutoff explicit filters. Second, the convection of turbulence through nonuniform grids is explored. This introduces additional challenges due to so-called commutation error, or neglect of the commutator of the filtering and differentiation operators. We employ a multiscale asymptotic analysis to investigate the characteristics of the commutator. Further, we show how commutation error manifests in simulation and demonstrate its impact on the convection of homogeneous isotropic turbulence through a coarsening grid. A connection is made between the commutation error and the propagation properties of the underlying numerics. A framework for modeling this commutator is proposed that accounts for properties of the discretization. The forcing of turbulence convecting through a refining grid is also explored and a formulation based on divergence-free wavelets is proposed. Results in isotropic turbulence suggest this formulation may be effective at energizing newly resolvable scales and therefore allowing for sharper grid transitions to finer resolved regions. There are several additional challenges to formulating more broadly applicable subgrid models for LES and we expect that the techniques developed here will also be useful for addressing these wide range of issues.