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And F ( . ) are restricted to the element K. Using bilinearity, we advocate the first equation as the method for computing the resolvable scales part U R of the solution Uh: (8) Residual-free bubbles 23 This equation can be viewed as the Galerkin method for the space V R plus a term which takes into account the unresolvable scales space Vu. In order to solve (8), we need to compute a(uu, VR). Now Uu can be obtained from the second equation of (6). , Uu is the solution of the variational problem (9) Note that the right-hand side involves the residual on the coarse scales.
The convective contribution to the global flux is treated implicitly by mimicking the upwinding of a scalar linear flux function while the rest of the flux is discretized in an explicit way. Spatial accuracy is ensured by allowing nonoscillatory polynomial reconstruction procedures, while time accuracy is attained by adopting a Runge-Kutta stepping scheme. The method can be considered naturally in the framework of the implicit-explicit (IMEX) schemes and the properties of the resulting operators are analysed using the properties of M-matrices.
Russo, A. (1997): b = f g. Comput. Methods Appl. Mech. Engrg. , Russo, A. (1998): Further considerations on residual-free bubbles for advective-diffusive equations. Comput. Methods Appl. Mech. Engrg. 166, 25-33  Brezzi, E, Russo, A. (1994): Choosing bubbles for advection-diffusion problems. Math. Models Methods Appl. Sci. R. (1982): Streamline upwindlPetrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible NavierStokes equations. Comput. Methods Appl.