05-04
A Two-Level Variational Multiscale Method for Convection-Diffusion Equations
by John, Volker; Kaya, Songul; Layton, William
Preprint series: 05-04, Preprints
- MSC:
- 65M60 Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods
Abstract: 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.
Keywords: convection-dominated convection-diffusion equation, variational multiscale method, two-level method, efficient implementation
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