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A two-level variational multiscale method for convection-dominated convection-diffusion equations. (English) Zbl 1124.76028
Summary: This paper studies the error in, the efficient implementation of and time stepping methods for a variational multiscale method (VMS) for solving convection-dominated problems. The VMS studied uses a fine mesh \(C^{0}\) finite element space \(X^{h}\) to approximate the concentration and a coarse mesh discontinuous vector finite element space \(L^{H}\) for the large scales of the flux in the two scale discretization. Our tests show that these choices lead to an efficient VMS whose complexity is further reduced if a (locally) \(L^{2}\)-orthogonal basis for \(L^{H}\) is used. A fully implicit and a semi-implicit treatment of the terms which link effects across scales are tested and compared. The semi-implicit VMS was much more efficient. The observed global accuracy of the most straightforward VMS implementation was much better than the artificial diffusion stabilization and comparable to a streamline-diffusion finite element method in our tests.

76M10 Finite element methods applied to problems in fluid mechanics
76M30 Variational methods applied to problems in fluid mechanics
76R99 Diffusion and convection
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