A hybrid mixed finite element scheme for the compressible Navier-Stokes equations and adjoint-based error control for target functionals

摘要

The importance of computer-based modeling in technical and industrial use is evident. Especially the aerodynamic industry has a huge interest in reliable and robust methods for accurately computing aerodynamic flows. These flows are in general described by partial differential equations of mixed elliptic-hyperbolic type, and as it is usually not possible to solve these equations exactly, one has to rely on numerical methods. Industrial state-of-the-art codes are usually based on low-order approximations, meaning that the underlying algorithm has a low order of consistency with the partial differential equation. This results in very robust methods that have become extremely efficient and therefore extremely popular. However, it is assumed that methods having a higher order of consistency can outperform the before-mentioned methods. We thus investigate in this work (known) high-order discretization schemes such as Discontinuous Galerkin Finite Elements, and we present a new discretization method for the Navier-Stokes equations in the framework of Hybrid Mixed Methods. The newly developed method is a combination of well-established methods for the diffusive (elliptic) and the convective (hyperbolic) part. Another aspect of numerical computation is the following: The practitioner's interest is often not in the quality of a numerical solution per se, but in a few derived quantities J_i, the so-called target functionals. The error in these quantities can be approximated by an adjoint procedure, meaning that one solves an additional, linear equation to obtain a weight z that relates the residual of the discrete equations to the error. The resulting expression can be used for grid adaptation, meaning to optimize the given underlying grid with respect to the accuracy of the target functional. A property of a discretization that is very important in this context, is adjoint consistency. This property ensures that the discretization can also be used to compute an approximation z_h to z. We therefore analyze existing Discontinuous Galerkin methods and the newly developed Hybrid Mixed method with respect to this property, and show that they are in fact adjoint consistent. The adjoint idea is in principle based on a first order Taylor expansion. It is evident that this requires a high degree of smoothness, which is usually not available in solutions to aerodynamic flows. To address that problem, we present a new mathematical framework for the investigation of the adjoint methodology in the case where the flow is non-smooth, and we give numerical evidence on how to actually compute the adjoint.