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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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