×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI