ON THE CONVERGENCE OF A DUAL-PRIMAL SUBSTRUCTURING METHOD. January 2000

ON THE CONVERGENCE OF A DUAL-PRIMAL SUBSTRUCTURING METHOD JAN MANDEL AND RADEK TEZAUR January 2000 Abstract In the Dual-Primal FETI method, introduced by Farhat et al [5], the domain is decomposed into non-overlapping subdomains, but the degrees of freedom on crosspoints remain common to all subdomains adjacent to the crosspoint The continuity of the remaining degrees of freedom on subdomain interfaces is enforced by Lagrange multipliers and all degrees of freedom are eliminated The resulting dual problem is solved by preconditioned conjugate gradients We give an algebraic bound on the condition number, assuming only a single inequality in discrete norms, and use the algebraic bound to show that the condition number is bounded by C(1 + log 2 (H/h)) for both second and fourth order elliptic selfadjoint problems discretized by conforming finite elements, as well as for a wide class of finite elements for the Reissner-Mindlin plate model 1 Introduction This article is concerned with convergence bounds for an iterative method for the parallel solution of symmetric, positive definite systems of linear equations that arise from elliptic boundary value problems discretized by finite elements The original Finite Element Tearing and Interconnecting method (FETI) was proposed by Farhat and Roux [9] The FETI method consists of decomposing the domain into non-overlapping subdomains, enforcing that the corresponding degrees of freedom on subdomain interfaces coincide by Lagrange multipliers, and eliminating all degrees of freedom, leaving a dual system for the Lagrange multipliers The dual system was solved by preconditioned conjugate gradients with a diagonal preconditioner Evaluation of the dual operator involves the solution of independent Neumann problems in all subdomains, and of a small system of equations for the nullspace component Farhat, Mandel, and Roux [8] recognized that this system for the nullspace components plays the role of a coarse problem that facilitates global exchange of information between the subdomains, causing the condition to be bounded as the number of subdomains increases They also replaced the diagonal preconditioner by a block preconditioner with the solution of independent Dirichlet problems in each subdomain and observed numerically that this Dirichlet preconditioner results in a very slow growth of the condition number with subdomain size Mandel and Tezaur [13] proved that the condition number grows at most as log 3 (H/h), where H is subdomain size and h is element size, both in 2D and 3D Tezaur [19] proved that a method by Park, Justino, and Felippa [16] is equivalent to the method of [9] with a special choice of the constraint matrices, and proved the log 2 (H/h) bound for this variant For further comparison, see Rixen et al [18] Klawonn and Widlund [10] have used preconditioned conjugate residuals to solve a saddle problem keeping both the original degrees of freedom and the Lagrange multipliers, and obtained the asymptotic bound log 2 (H/h) using an extension of the theory of [13] The saddle point approach has the advantage that approximate solvers can be used for both the Neumann and the Dirichlet subdomain problems, at the cost This work was supported by NSF grant ECS-9725504 and by ONR grant N-00014-95-1-0663 Department of Mathematics, University of Colorado at Denver, Denver, CO 80217-3364, USA, and Department of Aerospace Engineering Sciences, University of Colorado at Boulder, Boulder, CO 80309-0429, USA Email: jmandel@coloradoedu Center for Aerospace Structures and Department of Aerospace Engineering Sciences, University of Colorado at Boulder, Boulder, CO 80309-0429, USA Email: rtezaur@coloradoedu 1

Report Documentation Page Form Approved OMB No 0704-0188 Public reporting burden for the collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, Arlington VA 22202-4302 Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to a penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number 1 REPORT DATE JAN 2000 2 REPORT TYPE 3 DATES COVERED - 4 TITLE AND SUBTITLE ON THE CONVERGENCE OF A DUAL-PRIMAL SUBSTRUCTURING METHOD 5a CONTRACT NUMBER 5b GRANT NUMBER 5c PROGRAM ELEMENT NUMBER 6 AUTHOR(S) 5d PROJECT NUMBER 5e TASK NUMBER 5f WORK UNIT NUMBER 7 PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) Office of Naval Research,875 North Randolph Streeet,Arlington,VA,22203-1995 8 PERFORMING ORGANIZATION REPORT NUMBER 9 SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10 SPONSOR/MONITOR S ACRONYM(S) 12 DISTRIBUTION/AVAILABILITY STATEMENT Approved for public release; distribution unlimited 13 SUPPLEMENTARY NOTES 11 SPONSOR/MONITOR S REPORT NUMBER(S) 14 ABSTRACT In the Dual-Primal FETI method, introduced by Farhat et al [5], the domain is decomposed into non-overlapping subdomains, but the degrees of freedom on crosspoints remain common to all subdomains adjacent to the crosspoint The continuity of the remaining degrees of freedom on subdomain interfaces is enforced by Lagrange multipliers and all degrees of freedom are eliminated The resulting dual problem is solved by preconditioned conjugate gradients We give an algebraic bound on the condition number, assuming only a single inequality in discrete norms and use the algebraic bound to show that the condition number is bounded by C(1+log2(H=h)) for both second and fourth order elliptic selfadjoint problems discretized by conforming nite elements as well as for a wide class of nite elements for the Reissner-Mindlin plate model 15 SUBJECT TERMS 16 SECURITY CLASSIFICATION OF: 17 LIMITATION OF ABSTRACT a REPORT unclassified b ABSTRACT unclassified c THIS PAGE unclassified 18 NUMBER OF PAGES 12 19a NAME OF RESPONSIBLE PERSON Standard Form 298 (Rev 8-98) Prescribed by ANSI Std Z39-18

of of solving a larger indefinite problem instead of a small positive definite problem In [10], Klawonn and Widlund have proposed new preconditioners, proved uniform bounds for modifications of the method for the case of coefficient jumps, which include an earlier algorithm of Rixen and Farhat [17], and provided further theoretical insights The original method of [9, 8] does not converge well for plate and shell problems, and the existence and the form of the coarse space depend on the singularity of the subdomain matrices Therefore, Mandel, Tezaur, and Farhat [14] and Farhat, Mandel, and Chen [7] proposed to project the Lagrange multipliers in each iteration on an auxiliary space An auxiliary space chosen so that the corresponding primal solutions are continuous on crosspoints made it possible to prove that the condition number does not grow faster than log 3 (H/h) for plate problems [14], and fast convergence was observed for plate [7] as well as shell problems [4] This method is now called FETI-2 For related results for symmetric positive definite problems, see [6, 18] and references therein The subject of this paper is the Dual-Primal FETI method (FETI-DP), introduced by Farhat et al [5] This method enforces the continuity of the primal solution at crosspoints directly by the formulation of the dual problem: the degrees of freedom on a crosspoint remain common to all subdomains sharing the crosspoint and the continuity of the remaining degrees of freedom on the interfaces in enforced by Lagrange multipliers The degrees of freedom are then eliminated and the resulting dual problem for the Lagrange multipliers is solved by preconditioned conjugate gradients with a Dirichlet preconditioner Evaluating the dual operator involves the solution of independent subdomain problems with nonsingular matrices and of a coarse problem based on subdomain corners The advantage of this method is a simpler formulation than the methods of [14, 7], there is no need to solve problems with singular matrices, and the method was observed to be significantly faster in practice for 2D problems However, the design of a good 3D variant of the method is an open problem [5] In this paper, we prove that the condition number of the FETI-DP method with the Dirichlet preconditioner does not grow faster than log 2 (H/h) for both second order and fourth order problems in 2D By spectral equivalence, the result for fourth order problems extends to a large class of Reissner-Mindlin elements for plate bending as in [11, 12, 14] After few initial definitions, which are substantially different, the analysis is related to the analysis developed in [13, 14] Just as the formulation of the present method is simpler, the analysis is simpler and more elegant than in [13, 14] The paper is organized as follows The notation and assumptions are introduced in Section 2 In Section 3, we review the algorithm from [5] Section 4 gives the algebraic condition number estimates; these estimates apply to partitioning any symmetric positive definite system, not necessarily originating from partial differential equations Finally, in Section 5, we prove the polylogarithmic condition number bounds for two model problems of the second and fourth order 2 Domain partitioning, notation, and assumptions We are concerned with iterative solution of symmetric, positive definite linear algebraic systems Partitioning of the system is motivated as follows Let Ω be a domain in R 2 decomposed into N s non-overlapping subdomains Ω 1, Ω 2,, Ω Ns, and each of the subdomains be a union of some of the elements Let u s be the vector of degrees of freedom for the subdomain Ω s corresponding to a conforming finite element discretization of a second order elliptic problem or a fourth order plate bending problem defined on Ω Let K s and f s be the local stiffness matrix and the load vector associated with the subdomain Ω s We denote the edges of the subdomains by Γ st = Ω s Ω t Corners 2

are endpoints of edges The subdomain vectors are partitioned as u s u s i (21) = u s r, u s c where u s i are the values of the degrees of freedom in the subdomain interior, us c the values of the degrees of freedom at the corners of the subdomain, and u s r are the remaining values of the degrees of freedom, ie those located on the edges of the subdomains between the corners The subdomain matrices are partitioned accordingly, K s = Ks ii Kir s Kic s (22) Kri s Krr s Krc s Kci s Kcr s Kcc s We use the block notation u = u 1 u Ns and K = diag(k s ) = K 1 K Ns The block vectors u i, u c and u r of all internal, all corner, and all remainder degrees of freedom, respectively, are then defined similarly, u i = u 1 i u Ns i, u r = u 1 r u Ns r, and u c = u 1 c u Ns c Vectors of values of degrees of freedom on the whole Ω s and the corresponding block vectors will be written as [ ] v 1 vr,c s v s r,c [ ] = r vr vc s, v r,c =, and also v r,c = v c vr,c Ns Let the global to local map be a 0 1 block matrix L = L 1 L Ns That is, for a global vector of degrees of freedom u g, L s u g is the vector of corresponding degrees of freedom on Ω s The map L is introduced so that we can state the problem to be solved, (23) L T KLu g = L T f, independently of the solution algorithm Note that ImL is the space of all vectors u that are continuous across the subdomain interfaces and that L is of full column rank We assume that each matrix K s is symmetric positive semidefinite and that K s is positive definite on the subspace of vectors that are zero at subdomain corners, {u s u s c = 0} We also assume that the global stiffness matrix L T KL is positive definite, or, equivalently, that K is positive definite on ImL These assumptions are satisfied in the intended finite element applications 3

3 Formulation of the algorithm In this section, we review the algorithm proposed in [5] in a form suitable for our purposes The degrees of freedom from both sides of each edge Γ st should coincide, (31) u s r Γ st u t r Γ st = 0 In (31), each pair of subdomains {s, t} is taken only once, with the order (s, t) chosen arbitrarily We write the constraints (31) as B r u r = 0, B r = [B 1 r,, B Ns r ] Note that it follows immediately from the definition of B r that (32) and, for any edge Γ st, (33) B r B T r = 2I (B T r B r u r ) s Γ st = ±(u s r Γ st u t r Γ st) Let B c be a matrix with 0, 1 entries implementing the global-to-local map on subdomain corners That is, the equation u c = B c u g c, B c = B 1 c B Ns c determines the common values of the degrees of freedom on subdomain corners from a global vector u g c From the construction, the space of all vectors of degrees of freedom continuous across the interfaces can be written as, ImL = {u B r u r = 0, u c ImB c } The problem (23) is reformulated as the equivalent constrained minimization problem 1 2 ut Ku u T f min, subject to B r u r = 0 and u c = B c u g c for some u g c, which is in turn equivalent to finding the stationary point of the Lagrangean L(u i, u r, u g c, λ) = 1 u s 2 vt Kv v T f + u T r Br T λ, v = [v s ], v s i = u s r Bcu s g c Eliminating u s i, us r, and u g c from the Euler equations, we obtain a dual system of the form, cf [5, Eq (14)], (34) F λ = g, and solve it using the preconditioned conjugate gradients method with the preconditioner (35) where (36) M = B r S rr B T r, S rr = diag(s s rr), S s rr = K s rr K s rik s ii 1 K s ir For details of the implementation and numerical results, see [5] 4

4 Algebraic bounds In this section, we prove bounds on the condition number of the iterative method defined by Eqs (34) and (35) Denote u = u T u and, for a symmetric positive semidefinite matrix A, denote the induced matrix seminorm u A = u T Au = A 1/2 u If A is known to be positive definite, we write u A instead or u A, because the seminorm is then known to be a norm From the minimization property of the Schur complement, we immediately obtain the following lemma, which characterizes the bilinear form associated with the preconditioner Lemma 41 It holds that u T r S rr u r = min{v t Kv v r = u r, v c = 0} The next lemma gives a more specific description of the matrix of the dual equation (34) Lemma 42 It holds that F = B S 1 r Br T, where the positive definite matrix S is defined by (41) u T r Su r = min{v t Kv v r = u r, v c ImB c } Proof Let L(u r, λ) = min L(u i, u r, u g c, λ) u i,u g c Then, (42) L(u r, λ) = 1 2 ut r Su r + u T r B T r λ u T r h r, with some h r Minimizing over u r, we get u r = S 1 (h r B T r λ) Substituting u r into (42) and taking the variation over λ gives (34) with F = B r S 1 B T r We can now characterize the norm induced by the dual matrix F Lemma 43 It holds that (43) Proof From Lemma 42, λ T F λ = max v r 0 vr T Br T λ 2 v r 2 S λ T F λ = S 1/2 Br T λ 2 w T = max S r 1/2 Br T λ 2 w r 0 w r 2 The substitution w r = S 1/2 v r yields (43) Let V = R p be the space of Lagrange multipliers In this space, define the norm and also the dual norm, µ V = B T r µ Srr (44) λ V = max µ 0 µ T λ µ V Since V = ImB r, substituting µ = B r w r, we can rewrite (44) as (45) wr T Br T λ λ V = max B rw r 0 Br T B r w r Srr 5

The main result of this section is the following theorem, which gives a bound on the minimal and maximal eigenvalues of the preconditioned operator M F Theorem 44 If there exists a constant c 1 such that for all w r, then Br T B r w r 2 S rr c 1 w r 2 S, λ max (MF ) λ min (MF ) c 1 4 Proof The proof is based on a comparison of (45) and (43) Using Lemma 43, the substitution v r = B T r B r w r, and the property (32), we find that λ T vr T Br T λ 2 vr T Br T λ 2 wr T Br T λ 2 F λ = max v r 0 v r 2 S max 0 v r ImBr T v Br r 2 S = 4 max B rw r 0 Br T B r w r 2 S Since, by definition, B T r B r w r 2 S B T r B r w r 2 S rr, we conclude, comparing with (45), that (46) λ T F λ 4 λ 2 V On the other hand, from the assumption, we obtain (47) λ T wr T Br T λ 2 F λ = max w r 0 w r 2 S c 1 max B rw r 0 w T r B T r λ B T r B r w r Srr = c 1 λ 2 V Trivially from the definition of M and the norm in V, we have (48) µ 2 V = µ T Mµ Using (46), (47), and (48) in [13, Lemma 31] completes the proof We now show how to verify the assumption of Theorem 44 from inequalities of a form that is more usual in substructuring and easier to estimate for boundary value problems Denote by E s,t the operator that extends the vector of values of degrees of freedom on Γ st, not including corners, by zero entries to a vector of values of degrees of freedom on the whole Ω s, and let E s be the set of all indices of neighbors Ω t of the domain Ω s, with a common edge Γ st Denote by S s the Schur complement on Ω s obtained by eliminating the interior degrees of freedom of Ω s, ie, u st S s u s = min{v st K s v s v s r,c = u s r,c} Our estimate is based on an a-priori bound of the error of approximating a vector of interface degrees of freedom that is continuous across the corners by a vector that is continuous also across the edges In the applications in Sec 5, the approximating vector will be chosen as the natural interpolation on the edges from the corners Theorem 45 Suppose there is a constant c 2 such that for every w r,c, w c ImB c, there exists u r such that B r u r = 0 and, for all s and all t E s, E s,t (w i r u i r) 2 S s c 2 w i r,c 2 S i, i = s, t (49) 6

Then, λ max (MF ) λ min (MF ) c 2n e, where n e is the maximum number of the edges of any subdomain Proof Let w r be given and define w c to be the optimal corner degrees of freedom from the definition of S, cf (41) Then (410) N s w r 2 S = wr,c s 2 S s s=1 Let u s r be as in the assumption of the theorem Then B r u r = 0, and, consequently, B T r B r w r = B T r B r (w r u r ) Extending B T r B r w r by zero values of all corner degrees of freedom, we get using the definition of S rr, cf, (36), that (411) N s Br T B r w r 2 S rr = vr,c s 2 S s, s=1 where Using the definition of E s,t, we have v s r = ( B T r B r (w r u r ) ) s, v s c = 0 v s r,c = t E s E s,t v s r Hence, from the triangle inequality, and then using the property (33) of B r and the triangle inequality again, it follows that v s r,c S s t E s E s,t v s r,c S s t E s ( E s,t (w s r u s r) S s + E s,t (w t r u t r) S s) Squaring the the last inequality, using the inequality (a + b) 2 2(a 2 + b 2 ), and the a-priori bound (49) yields vr,c s 2 S 2c ( s 2 w s r,c 2 S + s wt r,c 2 ) S t t E s By the summation over the subdomains and using (411) and (410), we can conclude that N s Br T B r w r 2 S rr = vr,c s 2 S 4c N s 2n s e wr,c s 2 S = 4c 2n s e w r 2 S s=1 s=1 It remains to use Theorem 44 7

5 Applications In this section, we verify the assumption of Theorem 45 for two model problems The Sobolev seminorm, denoted by u m,p,x, is the L p (X) norm of the generalized derivatives of of order m 0 of the function u The Sobolev norm is then defined by u m,p,x = [ u 0,p,X,, u m,p,x ] l p Sobolev norms for noninteger m are defined by interpolation Cf, eg, [2, 15] for details and references In particular, u 0,p,X = u 0,p,X = u Lp (X), and, on the boundary Γ of a domain in R 2, u 2 1 2 Γ,2,Γ = u(x) u(y) 2 Γ x u 2 dx dy (51) where 51 A second order elliptic problem Consider the boundary value problem Au = g in Ω, u = 0 on Ω, Av = d i,j=1 x i ( α(x) v(x) ), x j with α(x) a measurable function such that 0 < α 0 α(x) α 1 ae in Ω We assume that model problem (51) is discretized using conforming P1 or Q1 elements and denote by Vh P 1 (Ω) the corresponding finite element space that satisfies the usual regularity and inverse properties [2] Let h denote the characteristic element size Assume that all functions in Vh P 1 (Ω) vanish on the boundary of Ω Assume that each subdomain is the union of some of the elements and denote the space of restrictions of functions from Vh P 1 P 1 (Ω) to subdomains by Vh (Ωs ) For every vector of degrees of freedom u s, denote by I P 1 u s the corresponding finite element function The trace of this function is determined by degrees of freedom on the boundary only and we often abuse the notation to define the trace from the boundary values only For simplicity, we assume that the subdomains Ω i, i = 1,, N s form a quasiregular triangulation of the domain Ω Denote the characteristic size of the subdomains by H Finally, let C denote a generic constant independent of h and H Theorem 51 For the model second order problem, it holds that ( λ max (MF ) λ min (MF ) C 1 + log H ) 2, h where the constant does not depend on H and h Proof We only need to verify the assumptions of Theorem 45 For a given w r,c, define u r,c by linear interpolation from w c ; that is, I P 1 u s r,c is linear on all edges of Ω s, and u c = w c Writing w r,c = (w r,c u r,c ) + u r,c, we get from [1, Lemma 35] that I P 1 E s,t (w s r u s r) 1 2,2, Ωs C (1 + log H h ) w i r,c S s Using the uniform equivalence of the seminorms I P 1 vr,c s 1 2,2, Ωs vs r,c S s, cf, eg, [1], and the uniform equivalence of the seminorms v 1 v 2,2, Ωs 1 2,2, Ωt for functions with support on the edge Γ st, which follows from the fact that the subdomains are shape regular, we get (49) with c 2 = C ( ) 2 1 + log H h Remark 52 The assumptions that the subdomains have straight edges can be relaxed to accommodate the case of only shape regular subdomains with edges that are not straight in a standard way by mapping from one or more reference subdomains 8

52 A fourth order problem Consider a biharmonic boundary value problem in a variational form: Find u H 2 0 (Ω) such that (52) a(u, v) = f(v), v H 2 0 (Ω), where a(u, v) = Ω 11 u 11 v + 12 u 12 v + 22 u 22 v, u, v H 2 0 (Ω) Let the model problem (52) be discretized by reduced HCT elements [2] We use the same assumptions on the decomposition as in Section 51; in particular, the subdomain edges are straight Let Vh HCT (Ω) be the finite element space of HCT elements satisfying the usual regularity and inverse properties and the essential boundary conditions Note that Vh HCT (Ω) C 1 (Ω) H0 2 (Ω) On each element, a function v in Vh HCT (Ω i ) is determined by the values v(a i ) and the values of its derivatives v x j (a i ), j = 1, 2, at the vertices of the element Denote by t, n, t, and n the tangential and normal directions, and the tangential and normal derivative, respectively The traces of functions from Vh HCT (Ω) on Ω are pairs of functions (u, v) such that u is piecewise cubic, n v is piecewise linear, and u and u are consistent, t u = t v We will abuse notation and write the trace functions simply as (u, u) or just u The space of HCT trace functions is denoted by Vh HCT ( Ω) Denote the finite element interpolation operator by I HCT As in Section 51, we abuse the notation by defining the trace of the interpolation from the boundary degrees of freedom and write, eg, I HCT u s r,c Vh HCT ( Ω s ) We adopt the convention that all functions are understood extended by zero outside of their stated domain; eg, u Γ is zero outside of Γ The following lemma gives a bound on the interpolation from subdomain corners to edges Lemma 53 For every w Vh HCT ( Ω s ), define u Vh HCT ( Ω s ) by HCT interpolation from the corners of Ω s to the edges; that is, u = v and u = w at the corners of Ω s, and, on each edge Γ st, u is a cubic polynomial and n u is linear Then ( (w u) Γ st) ( 1 C 1 + log H ) (53) w 2,2, Ωs 1 h,2, Ωs 2 Proof Denote v = w u Vh HCT ( Ω) Then v = 0 and v = 0 on the corners of Ω s, so from [12, Lemma 41], it follows that ( v Γ st) ( 2 1 v 2 2,2, Ωs 1 + C 1 + log H ) (54) v 2 2,2,Γst 0,,Γ h st Since u Γ st is from a space of dimension 6 and all norms on a finite dimensional space are equivalent, we have in the case when the length of Γ st is one that u 1 2,2,Γst u 1 2,2,Γst u 0,,Γ st Since u 1 and u 2,2,Γst 0,,Γst are invariant to stretching the edge, we get (55) u 1 2,2,Γst C u 0,,Γ st in the general case by scaling We will show at the end of the proof that (56) u 0,,Γ st 5 w 0,,Γ st 9

From the discrete Sobolev inequality as generalized by [12, Lemma 42], we have ( w 2 0,,Γ C 1 + log H ) ( w 2 st 1 + 1 ) (57) h 2,2,Γst H w 2 0,2,Γ st Combining (54), (56), and (57), we obtain ( v 2 1 C 1 + log H ) ( w 2 2,2, Ωs 1 + 1 ) (58) h 2,2, Ωs H w 2 0,2, Ω s First consider the case when Ω s Ω = Then (53) is invariant to adding a linear function to w because v is the error of linear interpolation of w and because this adds only a constant to w So, without loss of generality, let Ω s w = 0 Then by the Poincaré-Friedrichs inequality (59) 1 H y 2 0,2, Ω s C y 2 1 2,2, Ωs, proved first on a reference domain and then scaled [3, 20] Now (59) and (58) give (53) In the case when Ω s Ω, there are some essential boundary conditions on Ω s Since (53) has already been proved in the absence of essential boundary conditions on Ω s, it is sufficient to restrict (53) onto the subspace defined by the boundary conditions, noting that u satisfies the boundary conditions as well It remains to prove (56) The inequality is trivial for n u because the normal derivative is interpolated linearly between the corners Let u L be the linear function on the edge Γ st defined by the values of u on the corners Then, using the triangle inequality, (510) t u 0,,Γ st t (u u L ) 0,,Γ st + t u L 0,,Γ st By a simple computation, we see that the Hermite basis function φ, φ(0) = 0, φ (0) = 1, φ(1) = φ (1) = 0, φ a polynomial of order 3, attains the maximum of φ at 0 Mapping the interval (0, 1) on the edge Γ st and noting that u u L is zero at the endpoints x 1 and x 2 of Γ st, we get using the triangle inequality that (511) t (u u L ) 0,,Γ st t (u u L )(x 1 ) + t (u u L )(x 2 ) 2 t w 0,,Γ st + 2 t u L 0,,Γ st, because t u(x i ) = t w(x i ), i = 1, 2 From the mean value theorem and the fact that u L (x i ) = w(x i ), i = 1, 2, it follows that (512) t u L 0,,Γ st t w 0,,Γ st, hence (511) gives (513) t (u u L ) 0,,Γ st 4 t w 0,,Γ st Now (510), (512), and (513) give (56) Theorem 54 For the fourth order model problem, ( λ max (MF ) λ min (MF ) C 1 + log H ) 2, h where the constant does not depend on H and h 10

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