## "Finite Elemente. Theorie, schnelle Löser   und Anwendungen in der Elastizitästheorie"

Springer Verlag 2003.   ISBN: 3 540 00122 0

Solutions of selected problems are provided.

### Ergänzungen (E) und Korrecturen (C)

 S.55 (E) The stiffness matrix for the model problem was determined here in a node-oriented way. We note that the matrices are assembled in a different way in real-life computations, i.e. element-oriented. First, the contribution of each triangle (element) to the stiffness matrix is determined by doing the computation only for a master triangle (reference element). Finally the contributions of all triangles are added. S.64 (E) Ein anderes Macroelement ist das Powell-Sabin Element; vgl. Powell und Sabin [1977]. Piecewise quadratic approximations on triangles. ACM Trans. Software 3, 316-325 S.129+10 (C) Approximation von V(g) ... S.131 (C) 4.11 Satz. Die Voraussetzungen von Satz 4.3 seien erfüllt und a(v,v) sei nichtnegativ für alle v in X. S.140 (E) The FE solution of the Raviart-Thomas element is related to the solution of the nonconforming P_1 element. This was described by Marini [1985] and in a less obvious way by Arnold and Brezzi [1985] in Theorem 2.2. L.D.Marini [1985], An inexpensive method for the evaluation of the solution of the lowest order Raviart-Thomas mixed method. SIAM J. Numer. Anal. 22, 493-496 S.144+6 und 7 (C) ||τ||_{0,h}² ≤ ch ∑e ||τ||_{0,e}² + ... b(τ,v) ≥ c^{-1} |v|_{1,h} ||τ||_... [Exponenten korrigiert] S.157-5 (E) Specifically the following pair of subspaces of Xh, Mh is stable: X~h := { v in Xh, (div v,q)=0 for all q spanned by the functions in Fig. 40 d on macroelements}, M~h := { q in Mh, spanned by the functions in Fig. 40 a-c on macroelements}. The approximation property is not deteriorated since X~h contains X_2h. S.162+2 (C) Wenn das Gebiet nicht einfach zusammenhängend ist und Löcher enthält, sind zusätzliche Basisfunktionen mit nicht-lokalem Träger nötig. S.165 (E) The edge oriented residuals cannot be neglected; cf. C. Carstensen and R. Verfürth (1999). Edge residuals dominate a posteriori error estimates for low order finite element methods. SIAM J. Numer. Anal. 36, 1571 - 1587 S.168+4 (E) vgl. Ainsworth und Oden [2000]. M. Ainsworth and T.J. Oden . A Posteriori Error Estimation in Finite Element Analysis. Wiley, Chichester (2000). S.169 (8.31) (C) Der Suffix der letzten Norm ist 0,T'. S.183 (E) Literatur zur Ungleichung von Kantorovich: L.V. Kantorovich: Functional Analysis and Applied Mathematics [in Russian], Uspechi mat nauk, 3 (1948), 89-185 S.206 (E) In a paper by I. Schur [1917, p. 217] we find the submatrices that are now termed as Schur complements. I. Schur, Potenzreihen im Innern des Einheitskreises. J. Reine Angew. Math. 147, 205-232 (1917) S.236 (E) Beweis von Satz 3.5. Man beachte, dass die Funktion ρ → ρ² + a(1 - ρ²) auf [0,1] nichtfallend ist, sofern 0 < a =< 1 gilt. S.245 (E) H.A.Schwarz, Vierteljahresschrift Naturforsch. Ges. Zürich, 15, 272-286 (1870) S.263 (1.5) (C) Eij = 1/2(...) + 1/2 sumk (d uk/d xi)(d uk/d xj)   Man beachte die Indizes in der Doppelsumme S.285 (E) Although it is hard to find genuine elements for the Hu-Washizu principle, the principle is often used as a point of departure; an example is the EAS method by Simo and Rifai [1990]. S.291 (3.45)-1 (C) Xh cap ker B = {0} S.292 (E) The method of Simo and Rifai [1990] avoids volume locking of the displacements and the strains, but the volumetric part of the stresses may still suffer from locking. D. Braess, C. Carstensen and B.D. Reddy [2004] Uniform convergence and a posteriori error estimators for the enhanced strain finite element method. Numer. Math. 96, 461-479. S.300 (E) The variational formulation (4.8) is appropriate for nearly incompressible material. In this context it is crucial that the ellipticity constant can be bounded uniformly in the Lamé constant lambda. This follows from ||tr σ|| ≤ C (||σ_deviatoric|| + ||div σ||) A proof of this inequality can be found in the book by Brezzi and Fortin [1991] p.161. S.325+6 (C) Pih : ... [Streiche den Exponnten und das Gleichheitszeichen.] S.325-8 (E) Another mesh-dependent norm which less conceals the connection with H-1(div) was introduced by Carstensen and Schöberl [2000]. Residual-Based a posteriori error estimate for a mixed Reissner--Mindlin plate finite element method. SFB-Report No. 00-31, University of Linz
3. März 2007