A Least-Squares Formulation of the Moving Discontinuous Galerkin Finite Element Method with Interface Condition Enforcement

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.