## "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)

 p.58 (E) 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. p.61 (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. p.72+9 (C) ... It is concealed in the DKT element VI.5.3. p.74 (E) Due to Problem 5.14 a piecewise polynomial vector-valued function belongs to H(div), if its normal components are continuous at interelement boundaries. Similarly due to Problm VI.4.8 it belongs to H(rot), if its tangential components are continuous. The reader is warned for a wrong conclusion on the Sobolev spaces. The relatinn H(div) ∩ H(rot) = H1 is not always true. p.103 (C) Clearly, we get the same stiffness matrix for the triangulations shown in Figs. 27a and 27b. p.109 (E) The mesh-dependent norm || . ||h is the broken H1 semi-norm. Often a broken Sobolev norm is an appropriate mesh-dependent norm; c.f., (II.6.1). Jump terms as in the treatment of the Raviart-Thomas element in §5 are possibly added. p.111-3 (C) The remark "This corresponds with the practical observation that nonconforming elements are much more sensitive to near singularities i.e., to the appearance of large H² norms" is not true. Theorem III.5.6 says that the errors of the conforming P1 element and of the nonconforming P1 element are comparable independently of the regularity. p.116 (E) Problem 1.13. There seems to be a natural nonconforming P2 element of Crouzeix-Raviart type. Let 0 < α < ½. The natural nodes on the unit triangle are z1=(α,0), z2=(1-α,0), z3=(1-α,α), z4=(α,1-α), z5=(0,1-&alpha), z6=(0,&alpha). Show that the interpolation by quadratic polynomials at these points, however, is not well posed, since p(z1) - p(z2) + p(z3) - p(z4) + p(z5) - p(z6) = 0. p.125+3 (E) The importance of the following theorem for the finite element theory was pointed out by Babuška [1971]; see also Babuška and Aziz [1972]. The theorem is also found under the name Banach-Nečas-Babuška-Theorem. It can be traced back to Nečas [1962], Nirenberg and possibly to Banach; cf. Babuška [1971]. p.148 (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 and Burchard pointed out that the hypercircle method can be traced back to Friedrichs and Trefftz. 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. p.152 (E) Theorem 5.6. An error estimate for the finite element solution with the nonconforming P1 element can be included in the comparison. It was improved by C. Carstensen and D. Petersheim. D. Braess (2009). Calcolo 46, 149-155.   C. Carstensen and D. Petersheim (2011) p.152+17/18 (C) An index h is misplaced. On the left-hand side always: σh,     on the right-hand side always: σ p.160+1 (C) The inequality (6.12) is called Nečas inequality. p.160 (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 = ∞. p.177 (8.26) (C) c-1 he|| ... p.180 (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? p.181 (E) The term c||f-fh|| that arises from the data oscillation can even be added by the Pythagorean rule instead of the triangle inequality; see Ainsworth cited below. p.183 (E) Theorem 9.2 and Algorithm 9.3 can be found with different notation in Destuynder, P. and Métivet, B. (1999): Explicit error bounds in a conforming finite element method. Math. Comp. 68, 1379-1396 p.184 (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 p.185 (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. p.185 (E) Remark. The Poincaré Lemma for the inversion of the divergence in 2-space does not help here. To understand this determine the divergence of  ½ ∫01 txf(tx)dt. p.190 (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.216+2 (C) -1/ai ≤ bi, ci < 0 p.263-5 (C) ||uk+1-u||² ≤ ||uk+(αk²-1) v^ -u||² (v^ means hat v) p.293 (C) Usually the boundary integral in (3.1) is included with g replaced by -g, and all terms with g in this section carry the opposite sign. p.296  (3.14) (C) b(τ,v) = (div &tau,v)0 p.299 (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. p.299 (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. p.301+10 (C) div &sigma = – f p.301 (E) The minimization of the stress energy is also denoted as principle of Castigliano. p.311 (3.54) (C) The denominator is 4 since the correct denominators in the formulas above are 12. p.313+5 (C) Equation label (3.61) is missing. p.313+13 (C) γh=t-2Qh(w'h-&thetah) p.313 (3.62)+2 (C) The integrand of the second term is [Qh(wh'-θh)]². An index h is missing in the third term. p.314-3 (C) functions v of the form (3.16) pp.315-325 (C) Identify (x,y,z) with (x1,x2,x3). p.323 (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) p.326+12 (C) The Babuska paradox holds for simply-supported Kirchhoff plates. p.329+12 (C) cf. Problem II.5.16. p.335+10 (C) ... as we did in going from (3.29) to (3.31) and ... p.336 (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) p.351+5 (C) Braess, D. and Schöberl, J. (2006): Equilibrated residual error estimator for edge elements. Math. Comp. 77, 651-672 (2008)
April 5, 2016

## 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.