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Stabilized finite element methods. I.: Application to the advective- diffusive model. (English) Zbl 0759.76040
Summary: Some stabilized finite element methods for the Stokes problem are reviewed. The Douglas-Wang approach [J. Douglas jun. and J. Wang, Math. Comput. 52, No. 186, 495-508 (1989; Zbl 0669.76051)] confirms better stability features for high order interpolations. Next, the advective-diffusive model is approximated in the light of various stabilized methods, a global convergence analysis is presented and numerical experiments are performed. Biquadratic elements produce better numerical results under all stabilized methods examined. The design of the stability parameter is confirmed to be a crucial ingredient for simulating the advective-diffusive model, and some improved possibilities are suggested. Combinations of these methodologies are given in the conclusions and will be examined in detail in the sequel to this paper applied to the incompressible Navier-Stokes equations.

MSC:
76M10 Finite element methods applied to problems in fluid mechanics
76D07 Stokes and related (Oseen, etc.) flows
76R99 Diffusion and convection
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