A MULTISCALE FORMULATION OF THE DISCONTINUOUS PETROV-GALERKIN METHOD FOR ADVECTIVE-DIFFUSIVE PROBLEMS CARLO L. BOTTASSO † , STEFANO MICHELETTI ‡ , AND RICCARDO SACCO ‡ Abstract. We consider the Discontinuous Petrov-Galerkin method for the advection-diffusion model problem, and we investigate the application of the variational multiscale method to this formulation. We show the exact modeling of the fine scale modes at the element level for the linear case, and we discuss the approximate modeling both in the linear and in the non-linear cases. Furthermore, we highlight the existing link between this multiscale formulation and the p version of the finite element method. Numerical examples illustrate the behavior of the proposed scheme. Key words. Finite element method, mixed and hybrid methods, discontinuous Galerkin, Petrov- Galerkin, advection-diffusion problem, variational multiscale method AMS subject classifications. 65N12, 65N15, 65N30 1. Introduction and Motivation. We have recently introduced the Discon- tinuous Petrov-Galerkin (DPG) method for the solution of elliptic problems [5] and isotropic/anisotropic linear compressible and incompressible elasticity (Stokes prob- lem) [3, 8]. We propose in these pages some remarks on the application of the DPG method to the advection-diffusion model problem in one spatial dimension. This is a necessary step towards the extension of the method to the compressible and incom- pressible Navier-Stokes equations, our final goal. As shown in ref. [1], most Discontinuous Galerkin (DG) methods can be classified according to the selection of a specific expression for the “numerical fluxes” that are used for connecting neighboring elements. This expression, together with the choice of the functional spaces, effectively characterizes each method. The numerical fluxes are defined by expressing the element interface fields (e.g., edge variables in two spatial dimensions, or face variables in three spatial dimensions) as suitable averages of the internal fields for the elements sharing that interface. Slightly deviating from this philosophy, in the DPG method all unknown fields are approximated by internal and boundary variables, as it is usually done in mixed-hybrid formulations. The internal (mixed) variables are discontinuous across element interfaces as in other DG methods. However the boundary (hybrid) variables, while maintaining their classical role of connectors, are in this case treated as additional problem unknowns. Therefore, we do not have to select a recipe up-front in order to explicitly define the connectors in terms of the internal variables for formulating the method. This comes with a price to be paid, since it implies that the test and trial functions must now be chosen in different spaces, i.e. the resulting method is a mixed-hybrid Petrov-Galerkin one. It can be rigorously shown that this process yields higher order accurate interface unknowns (see [2, 5] and [7] for the analysis in one and two spatial dimensions, respectively). This behavior has also been experimentally observed in higher dimensional problems (see [5, 7]). It is well known that most numerical schemes fail to properly treat the finer † D. Guggenheim School of Aerospace Engineering, Georgia Institute of Technology, 270 Ferst Dr., 30332-0150 Atlanta GA, USA (carlo.bottasso@ae.gatech.edu). ‡ MOX–Modeling and Scientific Computing, Dipartimento di Matematica “F. Brioschi”, Po- litecnico di Milano, Via Bonardi 9, 20133 Milano, Italy (stefano.micheletti@mate.polimi.it, riccardo.sacco@mate.polimi.it). 1