Analysis of new augmented Lagrangian formulations for mixed finite element schemes
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.
scientific article; zbMATH DE number 1027742