  A least-squares formulation of the Moving Discontinuous Galerkin Finite
Element Method with Interface Condition Enforcement (LS-MDG-ICE) is presented.
This method combines MDG-ICE, which uses a weak formulation that separately
enforces a conservation law and the corresponding interface condition and
treats the discrete geometry as a variable, with the Discontinuous
Petrov-Galerkin (DPG) methodology of Demkowicz and Gopalakrishnan to
systematically generate optimal test functions from the trial spaces of both
the discrete flow field and discrete geometry. For inviscid flows, LS-MDG-ICE
detects and fits a priori unknown interfaces, including shocks. For
convection-dominated diffusion, LS-MDG-ICE resolves internal layers, e.g.,
viscous shocks, and boundary layers using anisotropic curvilinear
$r$-adaptivity in which high-order shape representations are anisotropically
adapted to accurately resolve the flow field. As such, LS-MDG-ICE solutions are
oscillation-free, regardless of the grid resolution and polynomial degree.
Finally, for both linear and nonlinear problems in one dimension, LS-MDG-ICE is
shown to achieve optimal convergence of the $L^2$ solution error with respect
to the exact solution when the discrete geometry is fixed and super-optimal
convergence when the discrete geometry is treated as a variable.
