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