Data-driven variational multiscale modeling of subgrid-scale effects in discontinuous Galerkin methods

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For large eddy simulation based on a Galerkin formulation, the variational multiscale (VMS) method provides a mathematically rigorous basis for the construction of closure models, decomposing the solution space into a coarse-scale (finite element) approximation and an infinite dimensional fine-scale complement. Discontinuous Galerkin (DG) methods are particularly suitable for flow simulations due to their robustness, conservation properties, and higher-order accuracy. DG methods, however, have been incompatible with the VMS framework established to date due to their discontinuous basis and associated variational flux terms. We recently introduced a new VMS-DG framework that reconciles the discontinuous Galerkin approach with the variational multiscale method, based on a specific VMS fine-scale closure function. Each fine-scale closure function emerges as the solution of a variational fine-scale problem, but in contrast to the established fine-scale Green’s function naturally accounts for contributions across discontinuities and avoids tedious convolution. Moreover, we demonstrated for the advection-diffusion equation that unlike in continuous Galerkin methods, fine-scale closure functions in DG discretizations exhibit a highly localized support. In this project, we leverage this foundation to develop a new data-driven subgrid-scale modeling methodology in a DG framework. It is based on three central hypotheses: (1) Due to the localization of fine-scale closure functions, DG methods enable their accurate element-local modeling as approximate solutions of the variational fine-scale problem, also for the (incompressible) Navier-Stokes equations. (2) The computational challenge of solving a very large number of such fine-scale problems during run-time can be tackled by computing fine-scale closure solutions via modern data-driven model order reduction technology. (3) Due to the consistent (residual-based) VMS closure formulation, the data-driven methodology remains naturally intertwined with the governing equations (i.e. the physics) and can thus appropriately represents subgrid-scale effects in large eddy simulations. Our research program involves the extension of our new VMS-DG framework to the incompressible Navier-Stokes equations, the derivation and investigation of two modeling variants for localized fine-scale closure functions in a DG context, and the development of a nonlinear DEIM-coupled reduced basis method and its data-driven calibration to enable their practical computation via extremely efficient solution of variational fine-scale problems. The feasibility of the data-driven approach, its computational efficiency, and the accuracy of large eddy simulations that can be achieved through the element-local subgrid-scale model are tested via well-established benchmark problems.