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Analysis of new augmented Lagrangian formulations for mixed finite element schemes. (English) Zbl 0874.65085
The authors analyse new augmented Lagrangian formulations for mixed finite element schemes which are useful to solve classical problems like the Stokes problem, the Laplace equation, the Reissner-Mindlin plate and shell problems. Such formulations are used to approximate solutions of minimization problems like: find $$u$$ minimizing the functional $F(v)= {\textstyle {\frac12}} a(v,v)-\langle f,v\rangle$ under the linear constraint $$Lu=0$$. They consist of
– introducing a Lagrange multiplier $$\mu$$ and on approximating the saddle-point $$(u,\lambda)$$ of the functional $G(v,\mu)= F(v)+\langle Lv,\mu\rangle;$ – replacing $$F$$ by $$\widetilde{F}$$, i.e., $\widetilde{F}(v)= F(v)+ {\textstyle {\frac\gamma2}}|Lv|^2,$ in order to prevent spurious modes.
The present contribution concerns the analysis of the augmented Lagrangian method when the parameter $$\gamma$$ depends on a power of the mesh-size $$h$$. It is proved that such a dependency involves in some cases an improvement of the error estimates.

##### MSC:
 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 65N15 Error bounds for boundary value problems involving PDEs 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation 35Q30 Navier-Stokes equations 74K20 Plates 35J25 Boundary value problems for second-order elliptic equations 74S05 Finite element methods applied to problems in solid mechanics 74K15 Membranes
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