## The Galerkin gradient least-squares method.(English)Zbl 0699.65077

The Galerkin gradient least-squares method introduced by T. J. R. Hughes and A. Brooks [Finite element methods for convection dominated flows, Winter ann. Meet. ASME, New York 1979, AMD Vol. 34, 19- 35 (1979; Zbl 0423.76076) and Proc. 3rd int. Conf. Finite elements in flow problems, Banff/Alberta 1980, Vol. II, 283-292 (1980; Zbl 0446.76077)] has allowed to substantially improve the Galerkin approximations of some engineering problems like thin structures, incompressible media, fluid flows etc., where spurious oscillations, locking and other undesirable features appeared.
By noting that the direct application of such a method to axisymmetric shell problems cannot overcome the dependence on the shell thickness, the authors analyze in this paper a simpler scale model equation, i.e., a singular diffusion problem given by $$\sigma^ 2u-\epsilon^ 2\Delta u=f.$$ The direct applications of standard Galerkin method and Galerkin/least-squares method do not overcome the spurious oscillations which appear when the ratio $$\epsilon^ 3/\sigma^ 2$$ becomes too small. To cure this shortcoming, the authors add to the Galerkin method a least-squares form of the gradient of the Euler-Lagrange equation; in this way, they obtain stability in the $$H^ 1$$ seminorm instead of $$L_ 2$$-stability in previous works. This new method is analyzed in detail for the one-dimensional model and then generalized to the multi-dimensional equation. Finally, numerical experiments confirm the good stability and accuracy of this method.

### MSC:

 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 35J25 Boundary value problems for second-order elliptic equations 74S05 Finite element methods applied to problems in solid mechanics

### Citations:

Zbl 0423.76076; Zbl 0446.76077
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### References:

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Van Leer, eds., Notes on Numerical Fluid Mechanics (Vieweg, Braunschweig, to appear). 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Franca, New mixed finite element methods, Ph.D. Thesis, Applied Mechanics Division, Stanford University, CA 94305, U.S.A. · Zbl 0651.65078 [28] Franca, L.P.; Hughes, T.J.R., Two classes of mixed finite element methods, Comput. methods appl. mech. engrg., 69, 89-129, (1988) · Zbl 0651.65078 [29] Franca, L.P.; Hughes, T.J.R.; Loula, A.F.D.; Miranda, I., A new family of stable elements for nearly incompressible elasticity based on a mixed Petrov-Galerkin finite element method, Numer. math., 53, 123-141, (1988) · Zbl 0656.73036 [30] Loula, A.F.D.; Franca, L.P.; Hughes, T.J.R.; Miranda, I., Stability, convergence and accuracy of a new finite element method for the circular arch problem, Comput. methods appl. mech. engrg., 63, 281-303, (1987) · Zbl 0607.73077 [31] Loula, A.F.D.; Hughes, T.J.R.; Franca, L.P.; Miranda, I., Mixed Petrov-Galerkin method for the Timoshenko beam, Comput. methods appl. mech. engrg., 63, 133-154, (1987) · Zbl 0607.73076 [32] Franca, L.P.; Loula, A.F.D., A new mixed finite element method for the Timoshenko beam problem, () · Zbl 0737.76044 [33] Hughes, T.J.R.; Franca, L.P., A mixed finite element formulation for Reissner-Mindlin plate theory: uniform convergence of all higher-order spaces, Comput. methods appl. mech. engrg., 67, 223-240, (1988) · Zbl 0611.73077 [34] A.F.D. Loula, I. Miranda, T.J.R. Hughes amd L.P. Franca, A successful mixed formulation for axisymmetric shell analysis employing discontinuous stress fields of the same order as the displacement field, in: Proc. Fourth Brazilian Symposium on Piping and Pressure Vessels 2 (Salvador, ABCM, Brazil) 581-599. 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