Large-Eddy Simulations (LES) and Direct Numerical Simulation (DNS) are applied to the analysis of a swirl burner operated with a lean methane–air mixture and experimentally studied by Meier et al. [19]. LES is performed for various mesh refinements, to study unsteady and coherent large-scale behavior and to validate the simulation tool from measurements, while DNS enables to gain insight into the flame structure and dynamics. The DNS features a 2.6 billion cells unstructured-mesh and a resolution of less than 100 microns, which is sufficient to capture all the turbulent scales and the major species of the flame brush; the unresolved species are taken into account thanks to a tabulated chemistry approach. In a second part of the paper, the DNS is filtered at several filter widths to estimate the prediction capabilities of modeling based on premixed flamelet and presumed probability density functions. The similarities and differences between spatially-filtered laminar and turbulent flames are discussed and a new sub-grid scale closure for premixed turbulent combustion is proposed, which preserves spectral properties of sub-filter flame length scales. All these simulations are performed with a solver specifically tailored for large-scale computations on massively parallel machines.