Goal‐oriented model reduction of parametrized nonlinear partial differential equations: Application to aerodynamics

Abstract

SummaryWe introduce a goal‐oriented model reduction framework for rapid and reliable solution of parametrized nonlinear partial differential equations with applications in aerodynamics. Our goal is to provide quantitative and automatic control of various sources of errors in model reduction. Our framework builds on the following ingredients: a discontinuous Galerkin finite element (FE) method, which provides stability for convection‐dominated problems; reduced basis (RB) spaces, which provide rapidly convergent approximations; the dual‐weighted residual method, which provides effective output error estimates for both the FE and RB approximations; output‐based adaptive RB snapshots; and the empirical quadrature procedure (EQP), which hyperreduces the primal residual, adjoint residual, and output forms to enable online‐efficient evaluations while providing quantitative control of hyperreduction errors. The framework constructs a reduced model which provides, for parameter values in the training set, output predictions that meet the user‐prescribed tolerance by controlling the FE, RB, and EQP errors; in addition, the reduced model equips, for any parameter value, the output prediction with an effective, online‐efficient error estimate. We demonstrate the framework for parametrized aerodynamics problems modeled by the Reynolds‐averaged Navier‐Stokes equations; reduced models provide over two orders of magnitude online computational reduction and sharp error estimates for three‐dimensional flows.