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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.

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