Consistent FE-analysis of elliptic variational inequalities

Economical mesh structures are of great interest when simulating physical processes using
the Finite Elemente Method. They are essential
for a fast calculation producing results of high accuracy. In case of restricted problems, many
a posteriori estimators which are the indicators for adaptive refinement turn out to be inconsistent
in areas where the restriction takes place. The effort of the subject matter is to develop a
method to overcome this problem by introducing saddle point
formulations and using the Lagrangian multiplier to balance gaps in the error estimations.
When dealing with sattle point problems there may arise the problem of unstable systems due to an
injured inf-sup-condition, especially in the discrete case.
We solve this problem using the Galerkin least squares method. In consequence
we get additional terms which also have to be taken into account when developing the a posteriori estimators.
To examine the general validity of this method we analyse problems of different type.
That means linear and nonlinear problems with linear or nonlinear restrictions in
the primal or dual variable, respectively.
In all cases, the resulting adaptive mesh structures
turn out out to be very efficient since they outline critical zones of the underlying problems
which is confirmed by numerical tests.