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Subgridscale stabilization of time-dependent convection dominated diffusive transport. (English) Zbl 1147.76039
Summary: In [W.J. Layton, Appl. Math. Comput. 133, No.1, 147–157 (2002; Zbl 1024.76026)], a variationally consistent eddy viscosity discretization is given for the stationary convection diffusion equation $u_ t-\varepsilon \Delta u+\vec b\cdot \nabla u+gu=f,\quad \vec x\in\Omega,\quad u=0,\;\vec x \text{ on }\partial\Omega$ . We further develop this discretization to include the time-dependent problem. We give comprehensive stability and error analysis of the semi-discrete case. We also state the stability and error results for the fully discrete algorithm with a Crank-Nicholson time discretization. The error bound is near optimal and independent of the diffusion coefficient, $$\varepsilon$$. Finally, we give guidance on optimal parameter selection for some common finite element spaces.

##### MSC:
 76M10 Finite element methods applied to problems in fluid mechanics 76R50 Diffusion 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 35B25 Singular perturbations in context of PDEs 35K20 Initial-boundary value problems for second-order parabolic equations 35K57 Reaction-diffusion equations
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