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In numerical simulations of Maxwell's equations for problems with disparate geometric scales, it is often advantageous to use grids of varying densities over different portions of the computational domain. In simulations involving structured finite-difference time-domain (FDTD) grids, this strategy is often referred as subgridding (SG). Although SG can lead to major computational savings, it is known to cause instabilities, spurious reflections, and other accuracy problems. In this paper, we introduce two strategies to combat these problems. First, we present an overlapped SG (OSG) approach combined with digital filters (in space). OSG can recover standard SG (SSG) schemes but it is based upon a more general, explicit separation between interpolation/decimation operations and the FDTD field update itself. This allows for a better classification of errors associated with the subgrid interface. More importantly, digital filters and phase matching techniques can be then employed to combat those errors. Second, we introduce SG with a domain overriding (SG-DO) strategy, consisting of overlapped (sub)grid regions that contain auxiliary (buffer) subdomains with perfectly matched layers (PML) to allow explicit control on the reflection and transmission properties at SG interfaces. We provide two-dimensional (2-D) numerical examples showing that residual errors from 2-D SG-DO FDTD simulations can be significantly reduced when compared to SSG schemes.