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

Cambridge University Press 2001.     ISBN: 0 521 01195 7

Solutions of selected problems are provided.
Some new material from the 2001 edition is listed.

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

 p.18+6 (C) The 9-point-stencil is the discretization of 1/12 [8*Δ u(x,y) + Δ u(x+h,y) + Δ u(x-h,y)   + Δ u(x,y+h) + Δ u(x,y-h)] p.58-9 (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. p.67 (E) Another macro-element is the Powell-Sabin element; see Powell and Sabin [1977]. Piecewise quadratic approximations on triangles. ACM Trans. Software 3, 316-325 p.125 (E) Theorem 3.6 became popular by the survey article by Babuška and Aziz, but it was originally provided by Nečas [1962].     J. Nečas (1962), Sur une méthode pour résoudre les équations aux dérivées partielles du type elliptique, voisine de la variationelle. Annali della Scuola Norm. Sup. Pisa 16, 305-326 p.132-9 (E) The inf-sup condition is also called Ladyšenskaya-Babuška-Brezzi condition (LBB-condition) since Ladyšenskaya provided an inequality for the divergence operator that is equivalent to the inf-sup condition for the Stokes problem. p.134 middle (E) If the bilinear form a(u,v) is not symmetric, the assumption (i) on the ellipticity in Theorem 4.3 has to be replaced by an inf-sup condition; cf. Brezzi and Fortin [1991, p.41]. p.138 (C) 4.11 Theorem. Suppose that the hypotheses of Theorem 4.3 are satisfied and that a(v,v) is non-negative for all v in X. p.144+16/17 (C) Add label (5.6) p.145-10/9 (C) More appropriate is -1/2 (σ,σ)0 → max! Then the primal and the dual prsigmaoblem attain the same value, and the non-optimal functions provide an inclusion. p.147 (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 p.149+13 (C) (5.13) is a direct consequence of Theorem 4.5. p.151+3 and 4 (C) ||τ||0,h ≤ ch ∑e||τ||0,e ... b(τ,v) ≥ c-1/2 |v|1,h ||τ||_... [exponents corrected] p.152-5 (C) Simo and Rifai [1990] p.156 (E) The inequalities in Theorem 6.3(2) are sometimes called Nečas' inequalities. p.163-5 (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 X2h. p.166, (7.14) (C) The superscript of the second π in the product is 1. p.168+15 (C) If the domain is not simply connected and there are holes in the domain, then additional basis functions with non-local support are required. p.171 (E) The edge oriented residuals cannot be neglected; cf. C. Carstensen und R. Verfürth (1999). Edge residuals dominate a posteriori error estimates for low order finite element methods. SIAM J. Numer. Anal. 36, 1571 - 1587 p.171 (E) A different approach to a posteriori estimators is found in the book by Neittaanmäki and Repin [2004]. It has the advantage that the greatest part of the estimator does not contain multiplicative constants as (8.12) and (8.13).     The application of the general theory to the Poisson equation is described. Let uh be the finite element solution with P1 elements. Suppose for the moment thsigmaat f=fh is piecewise constant. A function σ with div σ + fh = 0 can be determined with the Raviart-Thomas element on the same grid, and the Theorem of Prager and Synge (Theorem 5.1) provides the desired error bound for the solution to fh. For getting the difference to the solution of the original problem, the data oscillation f-fh is treated as it is done with residual estimators.     Moreover, in practice the solution for the Raviart-Thomas element is only approximated. The technique is the equilibrium calibration that is known from a posteriori estimators based on local Neumann problems; see Ainsworth and Oden [2000].     Neittaanmäki, P. and Repin, S. (2004), Reliable Methods for Computer Simulation. Error Cosigmantrol and A Posteriori Estimates. Elsevier, Amsterdam p.172 bottom (E) The general procedure that led to Theorem 8.1 can be described as follows. An isomorphism L: H → H-1 was associated to the given variational problem in section 3 and u-uh = L-1 l. The representation (8.16) of the H-1 function l helps to establish computable bounds of || l ||-1. - Residual error estimators can be also derived for other finite element discretizations in a similar manner. p.174+13 (E) See also Ainsworth and Oden [2000].     M. Ainsworth and T.J. Oden, A Posteriori Error Estimation in Finite Element Analysis. Wiley, Chichester (2000). p.175 (8.31) (C) The suffix of the last norm is 0,T'. p.189 (E) A reference to the Kantorovich inequality is:     L.V. Kantorovich: Functional Analysis and Applied Mathematics [in Russian], Uspechi mat nauk, 3 (1948), 89-185 p.200 (E) Problem 3.14. Let c>3 and define the symmetric 2×2 matrices A and B a11 = 2, a21 = c, a22 = 2c2, b11 = 3, a21 = 0, b22 = 3c2. Show that A ≤ B, but A2 ≤ B2 does not hold. p.212 (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) p.244 (E) Proof of Theorem 3.5. Note that the function ρ → ρ + a(1 - ρ) is nondecreasing on [0,1] whenever 0 < a ≤ 1. p.250 (E) A clear advantage has the cascadic version of multigrid algoritms for treating variational inequalities since the return to coarser grids is more involved there; see Blum, Braess and Suttmeier [2004], A cascadic multigrid algorithm for variational inequalities. Computing and Visualization in Science 7, 153-157 (2004) p.253 (E) H.A.Schwarz, Vierteljahresschrift Naturforsch. Ges. Zürich, 15, 272-286 (1870) p.254 (5.4)+2 (C) Since uk is constructed by a minimization in the subspace W,  we have Drop words in between. p.254-6 (C) Replace the factor α by its square in the definition of the test function. It induces obvious changes only in the intermediate expressions of the following formula. p.271 (1.5) (C) Eij = 1/2(...) + 1/2 sumk (d uk/d xi)(d uk/d xj)   note the indices in the sum p.293 (E) As was pointed out by Felippa [2000], the notation de Veubeke-Hu-Washizu principle would be more appropriate than the notation with two names since the three-field formulation can alreaddy be found in de Veubeke [1951].     Felippa, C.A. (2000), On the original publication of the general canonical functional of linear elasticity. J. Appl. Mech. 67, 217-219     Fraeijs de Veubeke, B.M. (1951), Diffusion des inconnues hyperstatique dans les voilure élongeron couplé.sigma Bull. Serv. Technique de L'Aéonautique No. 24, Imprimerie Marcel Hayez, Bruxelles, 56 p. p.294 (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]. p.296 (3.33) (C) ||u||2 + λ||div u||1 ≤ c ||f||0     [drop the 'h'] p.297-5 (C) Let the mapping divh : Xh → L2 p.298+4,+9,+10 (C) (3.34) → (3.33) (3.35) → (3.37) (3.37) → (3.39) (3.39) → (3.41) p.299 (3.45)-1 (C) Xh ∩ ker B = {0} p.301 (C) Remark 3.12.  3. Simo and Rifai [1990] extend the space in the spirit of Remark III.5.7 ... p.301 (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. p.309 (E) The variational formulation (4.8) is appropriate for nearly incompressible material. In this context it is crucial that the ellipticity constant can sigmabe bounded uniformly in the Lamé constant λ. This follows from ||tr σ|| ≤ C (||σdeviatoric|| + ||div σ||) A proof of this inequality can be found in the book by Brezzi and Fortin [1991] p.161. p.333+9 (C) By analogy with the derivation of (6.2) from (5.7) p.333-2 (C) ... in front of the shear term in (6.20) has to be ... p.334-13 (C) Πh : ... [drop the exponent and '='] p.335 (E) Another mesh-dependent norm which less conceals the connection with H-1(div) was introduced by C. Carstensen and J. 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 p.345 (E) Wang, Junping [1994], New convergence estimates for multilevel algorithms for finite-element approximations. J. Comput. Appl. Math. 50, 593-604
January 18, 2007

### Some new material from the 2001 edition.

The introduction of finite element spaces in Chapter II, Section 5 is now focused such that all the ingredients of the formal definition at the end of this section are well motivated. There are more estimates for the interpolation of Clement type.

The general considerations of saddle point problems in Chapter III are augmented. The direct and converse theorems that are related to Fortin interpolation are presented now under a common aspect. Mixed methods are often connected with a softening of the energy functional that is wanted in some applications for good reasons. A different but equivalent variational formulation that has become popular in solid mechanics, is easily understood.

In Chapter IV only the (short) standard proof of the Kantorowitch inequality has been replaced by a shorter one. The monotonicity of the function t+1/t (t > 1) and Young's inequality are sufficient.

The multigrid theory requires less regularity assumptions if convergence with respect to the energy norm is considered. A quick introduction into that theory is now included, and multigrid algorithms are also considered in the framework of space decompositions.

Finite element computations in solid mechanics require often appropriate elements in order to avoid an effect called "locking" by engineers. From the mathematical point of view we have problems with a small parameter. Methods for treating nearly incompressible material serve as a model for positive results while negative results are easily described for a more general framework.