In this talk, we will present adaptive filtering approach for shock capturing in DSEM to address these issues. By formulating physical constraints such as positivity and a local minimum entropy principle as constraints on the discrete solution, the filter strength is computed via a simple scalar optimization problem requiring only element-local information. Under some basic assumptions on the properties of the numerical scheme, the filtered solution is guaranteed to satisfy these constraints, resulting in an efficient but robust method for resolving discontinuous features without the use of problem-dependent tunable parameters. Furthermore, the proposed filtering approach recovers the standard DSEM approach for smooth solutions, retaining the efficiency, accuracy, and geometric flexibility of the underlying scheme. The efficacy of the approach will be shown in numerical experiments on hyperbolic and mixed hyperbolic/parabolic conservation laws such as the Euler and Navier-Stokes equations for problems including extreme shocks, shock-vortex interactions, and complex compressible turbulent flows.