Dietrich Braess
The Two-Energies Principle and the Justification
of Plate Models
The two-energies principle is now well established for computing a posteriori error estimates of finite element solutions. Standard elements approximate the solution of a minimum problem, and mixed methods provide approximations for the complementary maximum problem.
The difference yields an efficient upper bound for the error measured in the energy norm.It is less known that the principle gave rise to an approach to the justification of plate models 60 years ago. It was forgotten, since there was a serious gapin the proof. The possibility of boundar layers is a handicap here as well as in the approach with power series in the thickness of the plate. Now there are appropriate tools for singularly perturbed partial differential equations. In particular, an estimate with a square root in the thickness shows that it was not a trivial task to close the gap.