|
Ergänzungen wurden z.T. ohne Übersetzng von der englischen
Ausgabe desselben Jahres übernommen.
| ||
| S.22+19 | (C) | Dabei ist Ah die im Algorithmus 3.1 ... [Text am Zeilenanfang streichen] |
| S.55-56 | (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. The stiffness matrix associated to the stencil (4.9) induces a quadratic form z'Az = Σi,j (zi-zj)2 where the sum runs over all pairs i,j that belong to neighbors on the horizontal or vertical lines of the grid in Fig. 9. |
| S.58 | (E) |
5.1 Definition (3):
Shape regularity may be formulated as a condition on the angles of the
triangles in a triangulation. It follows from Oswald's re-examination
of an example by Babuška and Aziz in 1976
that convergence may fail if the triangulation is not shape regular.
Thus shape regularity or a similar condition is required.
P. Oswald, Divergence of FEM: Babuška-Aziz triangulations revisited. Applications of Mathematics 60 (2015), 473-484. |
| S.71 | (E) | Laut Aufgabe 5.14 gehört ein stückweise polymomiales Vektorfeld zu H(div), wenn die Normalkomponenten an den Kanten stetig sind. Ebenso gehört es nach Aufgabe III.5.15 zu H(rot), wenn die tangentiellen Komponenten stetig sind. Vor einem Trugschluss zu Sobolev-Räumen sei deshalb gewarnt. Es gilt nicht immer H(div) ∩ H(rot)=H1. |
| S.103-11 | (E) |
Die gitterabhängige Norm || . ||h ist streng genommen die
gebrochene H1-Seminorm. Meistens wird als gitterabhängige Norm eine gebrochene Sobolev-Norm verwandt wie schon in (II.6.1). Oft kommen wie beim Raviart-Thoma-Element in §5 noch Sprungterme hinzu. |
| p.110 | (C) |
Problem 1.11.
The difference of the bilinear forms a(.,.) and a_h(.,.) is not coercive.
Therefore a stronger assumption is required: |a(u,v)-ah(u,v)| ≤ ε ||u||.||v|| Moreover assume that the two bilinear forms are coercive with the constant α. |
| S.110 | (E) |
Problem 1.13.
Man erwartet instinktiv, dass es ein nichtkonformes P2 Element vom
Crouzeix-Raviart Typ gibt. Sei 0 < α < ½.
Natürliche Knoten im Referenzdreieck sind die Randpunkte z1=(α,0), z2=(1-α,0), z3=(1-α,α), z4=(α,1-α), z5=(0,1-α), z6=(0,α). Man zeige, dass die Interpolation mit quadratischen Polynomen wegen p(z1) - p(z2) + p(z3) - p(z4) + p(z5) - p(z6) = 0 nicht immer lösbar ist. |
| S.119+3 | (E) | Die Bedeutung des folgenden Satzes für die Finite-Elemente-Theorie wurde von Babuška [1971] sowie von Babuška und Aziz [1972] S. 112 herausgestellt. Er trägt zuweilen den Namen Banach-Nečas-Babuška-Theorem. Das Resultat findet man schon bei Nečas [1962], Nirenberg und sicherlich auch bei Sobolev. |
| S.141 | (E) |
The theorem of Prager and Synge is also called the
two energy principle, where the left hand side
of (5.5)v is the complementary energy.
Aubin und Burchard wiesen darauf hin, dass sich das
Zwei-Energien-Prinzip bis zu Trefftz und Friedrichs
zurückverfolgen lässt.
The right hand side is sometimes replaced by 2J(v)+2Jc(σ), where J and Jc denote the direct and the complementary energy. It is not only used for a posteriori error estimates, but also for a justification of plate models; cf. the addition to p. 323. K.O. Friedrichs (1929): Ein Verfahren der Variationsrechnung das Minimum eines Integrals als das Maximum eines anderen Ausdrucks darzustellen. Ges. Wiss. Göttingen, Nachrichten Math. Phys. Kl. 13-20. E. Trefftz (1928): Konvergenz und Fehlerschätzung beim Ritzschen Verfahren. Math. Ann. 100, 503-521. |
| S.142 | (E) |
The FE solution of the Raviart-Thomas element is related
to the solution of the nonconforming P1 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.145 | (E) | Theorem 5.6.
An error estimate for the finite element solution
with the nonconforming P1 element can be included
in the comparison. D. Braess (2009). Calcolo 46, 149-155 |
| S.154 (6.12) | (E) |
The inequality ||q||0 ≤ c ||grad q||-1 (and therefore the inf-sup condition for the Stokes problem) does not hold on arbitrary domains. Consider the domain with a cusp {(x,y); 0 < y < x², 0 < x < 1} and the function q(x,y) = x-2. It follows from ∂q/∂x = -2 ∂/∂y (y x-3), ∂q/∂y = 0 and ||y x-3||0 < ∞ that grad q belongs to H-1, but ||q||0 = ∞. |
| S.168 (8.11) | (C) | Bei ||RT||0,T2 fehlt der Faktor hT2. Die Faktoren der richtigen Formel entsprechen (8.10). |
| S.171 (8.26) | (E) | c-1 he1/2... |
| S.174 | (E) | Problem 8.5 An a posteriori error estimator is to be checked. Since the exact solution is not available, an (approximate) reference solution is computed by using finite elements of higher order. Is the real error overestimated or underestimated if merely the error with respect to the reference solution is evaluated? |
| S.177 | (E) |
Theorem 9.2 and Algorithm 9.3 findet man mit anderen Bezeichnungen bei Destuynder, P. and Métivet, B. (1999): Explicit error bounds in a conforming finite element method. Math. Comp. 68, 1379-1396 |
| S.178 | (E) |
The a posteriori estimator in Theorem 9.4 provides not only
an estimator for the P1 element, but also for the
mixed method of Raviart-Thomas. It is efficient since the
error of the mixed method is not dominant; see Theorem 5.6.
The converse is required if an efficient estimator for the Raviart-Thomas element is wanted. For this purpose the construction via piecewise quadratic elements by Ainsworth proceeds in a quite different way. Ainsworth, M. (2008): A posteriori error estimation for lowest order Raviart Thomas mixed finite elements. SIAM J. Sci. Comput. 30, 189-204 |
| S.178/180 Satz 9.4 | (E) |
Ein Detail zur Auswertung der Divergenz von
σ = ∇ uh + σΔ mag hilfreich sein.
Es ist ∇uh stückweise konstant und die punktweise berechnete Divergenz veschwindet, so dass div σ =fh punktweise gilt. Nach (9.4) sind die Normalkomponenten von σ an den Elementgrenzen stetig. Also ist div σ ∈ H(div), und die Gleichung div σ =-fh gilt in H(div) und nicht nur punktweise. Der Satz von Prager und Synge liefert die Fehlerschranke für die Poisson-Gleichung mit der rechten Seite fh. Die Differenz f-fh wird über die Dreiecksungleichung erfasst. |
| S.179 | (E) |
Remark.
The Comparison Theorem III.5.6 for the conforming P1 element,
the nonconforming P1 element, and the Raviart-Thomas element
contains an a priori estimate. There is the question:
Why are tools from the theory of a posteriori estimates used for its proof?
The reason is that the theorem is true only modulo data oscillation,
and the latter has been introduced and understood in the framework
of a posteriori estimates.
There is another fact of a similar type. The proof of the lower bound (8.19) uses cubic (and quadratic) bubble functions. It follows that P4 elements yield a solution with an error that is smaller than the error for P1 elements multiplied by a factor smaller than 1, provided that we disregard terms arising from data oscillation. We cannot do it without this addition, since it is easy to construct a right-hand side of the elliptic equation such that the finite element solution with P4 elements is contained in the subset of P1 elements. Obviously, the distance to the P4 solution does not reflect the distance to the true solution in this case. The use of techniques from a posteriori estimates for the a priori analysis of plates has a different reason. |
| S.187 (1.12) | (C) | Die genannten Eigenwerte müssen verdoppelt werden. |
| S.188 | (E) |
For a first convergence proof of The Gauss-Seidel method see:
v. Mises, R. and Pollaczek-Geiringer, H. (1929): Praktische Verfahren der Gleichungsauflösung. ZAMM 9, 58-77, 152-164. |
| p.210 -9 | (C) | -1/ai ≤ bi, ci < 0. [Vorzeichen]. |
| p.256-2 | (C) |
||uk+1-u||²
≤ ||uk+(αk²-1) v^ -u||² (v^ bedeutet Hut v) |
| S.286 | (C) | Das Randintegral wird üblicherweise mit negativem Vorzeichen erfasst. |
| S.292 -11 | (C) | Kragarm in Abb. 58 |
| S.292 -11 | (E) |
The constant in Korn's inequality depends on the shape of the domain
if Neumann boundary conditions are given on a part of the domain.
An Example is the cantilever beam (Fig. 58) with 0 ≤ x1 ≤ l, 0 ≤ x2 ≤ t, and t/l very small. Consider the displacement v1 = -3x12x2, v2 = x13. Elementary calculations show that |v|1 is large when compared with ||ε(v)|| since ε12(v) = ε22(v) = 0. Note that ||rot v|| is also large. |
| S.292 -11 | (E) | The inf-sup condition for the Stokes problem implies Korn's inequality. The counterexample of a domain with a cusp shows that there is no implication in the converse direction. |
| S.293 -3 | (E) | The minimization of the stress energy is also denoted as principle of Castigliano. |
| S.300 | (E) | Bemerkung 4.2 (3) 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.317 | (E) |
Hypothesis H2: Verifications of the plate models have been
performed by using the theorem of Prager and Synge (p. 148).
It was done by Morgenstern [1959] for the Kirchhoff plate
and by Braess, Sauter, and Schwab [2009] for both plate models.
While the studies above refer to the displacement model,
Alessandrini et al [1999] investigated mixed methods
for the Mindlin-Reissner plate.
Although not explicitly stated, the results show
that Hypothesis H2 makes the plates stiffer than they are.
An ansatz with a quadratic term
u3 = w(x,y) + z² w2(x.y) is required in both models. The lack of the quadratic term was often partially compensated by shear correction factors. Shear correction factors are disregarded in the text. Morgenstern, D. (1959): Herleitung der Plattentheorie aus der dreidimensionalen Elastizitätstheorie. (German) Arch. Ration. Mech. Anal. 4, 145-152. Alessandrini, A.L., Arnold, D.N., Falk, R.S., and Madureira, A.L. (1999): Derivation and justification of plate models by variational methods. In "Plates and Shells, Quebec 1996", (M. Fortin, ed.), pp. 1-20. CRM Proceeding and Lecture Notes, vol. 21, American Mathematical Society, Providence, RI. Braess, D., Sauter, S. and Schwab, C. (2009): On the justification of plate models. J. Elasticity 103, 53-71 (2011) The theory is contained in the 5th German edition of this book. |
| S.320 +16 | (C) | Das Babuška Paradoxon gilt für die einfach eingespannte Platte. |
| S.330 | (E) |
For a more recent survey of plate elements see: Falk, R.S. (2008): Finite elements for Reissner-Mindlin plates. In "Mixed Finite Elements, Compatibility Conditions, and Applications", (D. Boffi and L. Gastaldi, eds.), pp. 195-232, Springer (2008) |