## "Finite Elements. Theory, Fast Solvers   and Applications in Solid Mechanics"

Cambridge University Press 2007.     ISBN: 0-521-70518-5

Solutions of selected problems are provided.

Some specialties and highlights of the book

## Supplements and Extensions (E) and Corrections (C)

September 9, 2022

## Specialties in the book on Finite Elements

A textbook on finite elements contains much material that is standard and can be found in many books on this topic. We want to point at some specialties that the student might not notice without a hint. Moreover, there are some items that look elementary at first glance, but are useful for understanding recent research.

We will list some of the specialties.
• Ch.II §3.
Point functionals are not bounded on the set of H1 functions in 2-space. This is related to functions with possible singularities and is elucidated by the impact on the construction of tents.

• Ch.II §6.
It follows from approximation properties of finite element spaces what form the corresponding best inverse properties can have.

• Ch.III §4.
A barrier in the theory of saddle point problems with penalty terms is presented and discussed. The theory is now probably complete. It shows, e.g., why mixed methods for the Mindlin-Reissner plate are more involved than expected.

• Ch.III §§5 and 9.
The theorem of Prager-Synge and a posteriori error estimates by the two energies principle are established. It is shown how cheap error bounds without unknown generic constants are evaluated for symmetric variational problems in divergence form.
Error bounds of this type for hp-elements have also better asymptotic properties than residual estimators.

• Ch.III §5.
Raviart-Thomas elements and conforming P1 elements refer to different variational concepts. Nevertheless, there is a relation between the discretization errors of the two methods.

• Ch.III §5.
The softening behavior of mixed methods is discussed. This property is important for problems in solid mechanics with a small parameter.

• Ch.III §6.
A proof of the inf-sup condition for the Stokes problem is beyond the scope of this book, but the equivalence with the operators grad and div is discussed. It is shown in Ch.VI that the inf-sup condition for the Stokes problem implies Korn's inequality.

• Ch.IV §2.
A very short proof is provided for the Kantorovitch inequality.

• Ch.V §3.
The convergence proof for the V-cycle is included.

• Ch.VI contains a short introduction into elasticity theory in order to have a self-consistent presentation of finite elements in solid mechanics.

• Ch.VI §3.
A framework for the mathematical treatment of locking phenomena is presented. The theory for the Timoshenko beam is now complete. The crucial inequality for nearly incompresssible material is provided without proof.

• Ch.VI §§5 and 6.
The Kirchhoff plate and the Reissner-Mindlin plate are supposed to satisfy the hypothesis that the displacement in the z-direction does not depend on z. The analysis of plate models by the two energies principle, however, shows that the hypothesis makes the plates too stiff. [This knowledge is newer than the present edition.]

• Ch.VI §6.
The treatment of the Reissner-Mindlin plate without Helmholtz decomposition can be understood as the correct treatment of a saddle point problem with a singularly perturbed penalty term.