Supplements and Extensions (E) and Corrections (C) to Dietrich Braess, Finite Elements ================================= The severe corrections are labelled by (C*) Corrections of the German Edition are labelled by (G) II.1 ====== p.27 (E) The notation FINITE ELEMENTE is due to Clough and his group p.32+10 (C) the image of the unit ball U is relatively compact [=S.31] (1.10)+1 (C) form a relatively compact subset ---------- II.2 ====== S. 38-11 (G) sogenannte -8 (G) Beweis II.3 ===== S. 47 3.4+1 (G) Lipschitz p. 49 middle (C) H^1 is isomorhic to the direct sum [=S.46] H^1_0 + trace(H^1) II.4 ====== p.55 (4.7) (E) The relation (4.7) is often denoted [=S.52] as Galerkin orthogonality II.5 ====== p.60 (5.1) (C*) The sum runs over i+k<=t [=S.57] p.64 (5.3)-2 (C) polynomial p of degree lower or equal t S. 62 (5.4)+1 (C) M^1_0 kalligraphisch II.6 ===== S. 73+18 (G) ||L|| anstatt ||Lv|| zweimal nach kleiner gleich II.7 ===== S. 85+7 (G) in den meisten F"allen sind die Finite-Elemente-N"aherungen p. 91-8 (C) ||P_h v|| <= ||P_h v - v|| + minus! III.1 ====== S. 99+15 (G) gitterabh"angige Normen S.100 (G) Die Klammern <,> sind an die Klammern f"ur duale Paarungen anzupassen p.102 (E) The theorem which is well known as [=S. 99] Second Lemma of Strang is due to A.Berger, R.Scott, and G.Strang: Approximate Boundary Conditions in the Finite Element Method. Symposia Mathematica 10 (1972), 295-313 p.102-3 (C) The algebraic sum V+S_h need not be a direct sum. [=S. 99] p.103 (1.6)+1 (C) a(...) is better replaced by a_h [=S.100] p.104-5 (C) Specify T as the domain in the integral following the sum [=S.101-5] p.105+12 (C) The norm in the middle has to be squared. [=S.102+11] III.3 ====== p.123 (C) Problem 3.10. H^2 is to be replaced by [=S.120] the its intersection with H^1_0. (E) Problem 3.13. Often one finds the variational [=S.120] problem (3.15)_h with the following special situation U=V and the bilinear form a(.,.) is symmetric and elliptic, but U_h not= V_h. In this case the stability is related to the projector onto V_h. Verify this for the simplest case: Let U_h and V_h be subspaces of a Hilbert space with dim U_h = dim V_h and, given the functional f, find u_h in U_h such that (u_h, v) = (f, v) for all v in V_h. Show that ||u_h|| le (1/alpha) ||f||, if ||P u_h|| ge alpha ||u_h|| holds for the orthogonal projector P:U_h -> V_h. Moreover, 1/alpha is the best possible constant. III.4 ======= p.130 (E) After Remark 4.6: [=S.127] The mapping in the proof u |--> v_h =: \Pi_h u is sometimes called Fortin's interpolation. Note that \Pi_h is anoperator with the properties required in Fortin's Criterion. Thus the existence of such an operator is also a necessary condition in 4.9. - Drop Problem 4.14. p.131 (E) After Fortin's Criterion: [=S.128] Note that the condition in Fortin's criterion can be checked without referring explicitly to the norm of the Lagrangian multipliers. This is an advantage when the space of the Lagrangian multipliers is equipped with an exotic norm, and is thus used e.g. when the Lagrangian multipliers belong to trace spaces. Note also that the converse stated in Problem 4.14 is strongly related to the approximation property in Remark 4.6. p.134 middle (E) Sometimes (4.19) with X=L_2 and M=H^1_0 [=S.131] is called a primal mixed method while (4.21) with X=H(div) and M=L_2 is called a dual mixed method. p.134 middle (E) The dual mixed variational problem is related to THEOREM of Prager and Synge. Let sigma in H(div) satisfy div sigma+f = 0 and v in H^1_0 (Omega). Then we have for the solution u of the Poisson equation |u-v|_1^2 + ||grad u-sigma||_0^2 = ||grad v-sigma||_0^2. PROPLEM 4.25. Prove the theorem of Prager and Synge by showing that grad(u-v) and grad u-sigma are orthogonal in L_2. p.135 (E) The mixed method has the effect of a softening of the bilinear form. This can be derived from the variational form (4.19)_h, i.e. the discrete analog of (4.19). Assume that X_h \subset H^1_0(Omega) and M_h \subset L_2(Omega). Set E_h=grad X_h. If M_h \supset E_h, then the first equation yields \sigma_h =grad u_h. This is the uninteresting case. If M_h does not contain E_h, then we can write E_h=M_h\oplus ~E_h. In particular we can assume that ~E_h is orthogonal to M_h. Let P_h:L_2 -> M_h be the orthogonal projector onto M_h. Then the first equation in (4.19)_h states \sigma_h = P_h (grad u_h) and the second equation of (4.19)_h just states that u_h is the solution of \int [P_h (grad u_h)]^2 dx - 2 \int f v_h dx -> min! So the bilinear form of the original form is relaxed. Here the relaxation is defined by the finite element space ~E_h. We note that in structural mechanics an equivalent concept was derived by Simo and Rifai [1998]; calledi "method of enhanced assumed strains". The defined the projection by specifying the complementary space ~E_h instead of M_h. III.7 ====== p.158-12 (C) Averages of the gradients=nabla u and not of Delta u [=S.152] are used for the estimator. p.160+11 (E) Add hint to II.6.9 for Cl'ements' interpolation. [=S.156] p.160 (E) Note that (7.16) and (7.19) can be understood [=S.156] as representations for the (-1)-norm of u-u_h. p.161+7 (E) Specifically if Delta u_h=0 piecewise, then the first term in degenerates to the a priori computable expression (f, w-I_h w)_0. Now we replace I_h w by P_h w, see (7.7) (f, w-P_h w)_0 and now we may subtract any element from S_h, i.e., we have (f-f_h, w-P_h w)_0 and it is clear that the term is a term of higher order. Moreover, the upper and lower bounds (7.12) and (7.13) become closer if we take only the contributions of the edges into eta_R and add ||f-f_h||_0 in the upper bound. p.163 (E) One might think of searching the finite element function which minimizes the error estimator (7.11) instead of minimizing the given variational functional. This does not make sense. If we have linear elements and want to solve the Poisson equation, then the only variable term is the term R_e with the jumps over the interelement boundaries. Obviously u_h=0 would minimize the jumps. [=S.159] IV.1 ===== p.169+6 (C*) The 2,2-entry in the last matrix has the positive sign [ 0 -0.1 ] [ 0 +0.4 ] [=S.165] p.171 (1.12) (C*) Check all factors [=S.167] p.172 (1.17) (C) A closing ')' is missing [=S.168] p.174 (E) Problem 1.16. Let A be a symmetric, positive definit [=S.170] matrix and M be defined by (1.20) with omega=1. Show that M-A is positive semidefinit. IV.4 ===== (G) Cholesky ist die richtige Schreibweise p.199 (E) Problem 4.14. Show that A