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Feedback Petrov-Galerkin methods for convection-dominated problems. (English) Zbl 0753.76093
Summary: The Petrov-Galerkin method introduced in [ibid. 68, No. 1, 83-95 (1988; Zbl 0626.76091)] is adaptively applied to convection-dominated problems. To this end a feedback function is created which increases or decreases the control of the gradient of the approximate solution. This leads to a method with good stability properties close to boundary layers and high accuracy where regular solutions do occur.

MSC:
76M10 Finite element methods applied to problems in fluid mechanics
76R99 Diffusion and convection
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[1] Brooks, A.N.; Hughes, T.J.R., Streamline upwind Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations, Comput. methods appl. mech. engrg., 32, 199-259, (1982) · Zbl 0497.76041
[2] Hughes, T.J.R.; Mallet, M.; Mizukami, A., A new finite element formulation for computational fluid dynamics: II. beyond SUPG, Comput. methods appl. mech. engrg., 54, 341-355, (1986) · Zbl 0622.76074
[3] do Carmo, E.G.; Galeão, A.C., A consistent formulation of the finite element to solve convective-diffuse transport problems, Rev. brasileira cienc. mec., 4, 309-340, (1986), (in Portuguese)
[4] Galeão, A.C.; Dutra do Carmo, E.G., A consistent approximate upwind Petrov-Galerkin formulation for convection-dominated problems, Comput. methods appl. mech. engrg., 68, 83-95, (1988) · Zbl 0626.76091
[5] Johnson, C.; Nävert, U.; Pitkäranta, J., Finite element methods for linear hyperbolic problems, Comput. methods appl. mech. engrg., 45, 285-312, (1984) · Zbl 0526.76087
[6] Dutra do Carmo, E.G.; Galeão, A.C., High order P-G finite elements for convection-dominated problems, (), 151-156
[7] Mizukami, A.; Hughes, T.J.R., A Petrov-Galerkin finite element method for convection-dominated flows: an accurate upwind technique for satisfying the maximum principle, Comput. methods appl. mech. engrg., 50, 181-193, (1985) · Zbl 0553.76075
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