Efficient hyperreduction of high-order discontinuous Galerkin methods: Element-wise and point-wise reduced quadrature formulations

We develop and assess projection-based model reduction methods for high-order discontinuous Galerkin (DG) discretizations of parametrized nonlinear partial differential equations with applications in aerodynamics. Our emphasis is on the choice of hyperreduction methods. We analyze computational complexity and use numerical examples to show that typical hyperreduction methods based on element-wise or index-wise sampling of the residual vector cannot effectively reduce high-order DG discretizations, with a large number of degrees of freedom and quadrature points per element and conversely a small number of elements. To overcome this limitation, we devise a hyperreduction method for high-order DG methods that samples individual quadrature points, instead of elements, to provide a finer decomposition of the residual. We compare the formulation, implementation, and computational complexity of the element-wise and point-wise formulations. We then numerically assess the two formulations using Euler flow over a shape-parametrized airfoil and laminar and turbulent Navier-Stokes flow over a three-dimensional aerodynamic body. In each case, we observe that the point-wise hyperreduced model reduces the computational time by several orders of magnitude relative to the high-order DG method and a few orders of magnitude relative to the element-wise hyperreduced model. In addition, the computational performance of the point-wise formulation does not deteriorate with the DG polynomial degree, unlike for the element-wise formulation.