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On necessary and sufficient conditions for finite element convergence ∗ 6 1 0 Vaćlav Kuˇcera 2 n Faculty of Mathematics and Physics, Charles University in Prague a J January 13, 2016 2 1 ] A Abstract N Inthispaperwederiveanecessaryconditionforfiniteelementmethod . (FEM) convergence in H1(Ω) as well as generalize known sufficient con- h ditions. We deal with the piecewise linear conforming FEM on trian- t a gular meshes for Poisson’s problem in 2D. In the first part, we prove a m necessary condition on the mesh geometry for O(hα) convergence in the H1(Ω)-seminormwithα∈[0,1]. Weprovethatcertainstructures,bands [ consisting of neighboring degenerating elements forming an alternating 1 pattern cannot be too long with respect to h. This is a generalization v of the Babuˇska-Aziz counterexample and represents the first nontrivial 2 necessary condition for finite element convergence. Using this condition 4 we construct several counterexamples to various convergence rates in the 9 FEM.Inthesecondpart,wegeneralizethemaximumangleandcircumra- 2 dius conditions for O(hα) convergence. We prove that the triangulations 0 . cancontainmanyelementsviolatingtheseconditionsaslongastheirmax- 1 imumanglevertexesaresufficientlyfarfromotherdegeneratingelements 0 ortheyformclustersofsufficientlysmallsize. Whileanecessaryandsuf- 6 ficient condition for O(hα) convergence in H1(Ω) remains unknown, the 1 gap between the derived conditions is small in special cases. : v i X 1 Introduction r a The finite element method (FEM) is perhaps the most popular and important general numerical method for the solution of partial differential equations. In its classical, simplest form, the space of piecewise linear, globally continuous functionsonagivenpartition(triangulation) ofthespatialdomainΩisused h T alongwithaweakformulationoftheequation. Inourcase,wewillbeconcerned with Poisson’s problem in 2D. ∗This work is a part of the research project P201/13/00522S of the Czech Science Foun- dation. V. Kuˇcera is currently a Fulbright visiting scholar at Brown University, Providence, RI,USA,supportedbytheJ.WilliamFulbrightCommissionintheCzechRepublic. 1 Much work has been devoted to the a priori error analysis of the FEM. Namely, the question arises, what is the necessary and sufficient condition for the convergence of the method when the meshes are refined, i.e. we have a h T system of triangulation . In the simplest case, we are interested in the energy norm, i.e. H1{(TΩh)}-he∈st(0im,h0a)tes of the error u U C(u)h, (1) 1 | − | ≤ where u and U are the exact and approximate solutions, respectively, h is the maximal diameter of elements from and C(u) is a constant independent of h T h. Typically, one is interested in deriving (1) for some larger class of functions, e.g. for all u H2(Ω). ∈ Historically, the first sufficient condition for (1) to hold is the so-called minimum angle condition derived independently in [13], [14]. This condition states that there should exist a constant γ such that for any triangulation 0 ,h (0,h ), and any triangle K we have h 0 h T ∈ ∈T 0<γ γ , (2) 0 K ≤ where γ is the minimum angle of K. This condition was later weakened K independently by [1], [2] and [7] to the maximum angle condition: there exists a constant α such that for any triangulation ,h (0,h ), and any triangle 0 h 0 T ∈ K we have h ∈T α α <π, (3) K 0 ≤ whereα isthemaximumangleofK. Finally, aconditionforconvergencewas K recently derived in [8] and [12], the circumradius condition: Let max R 0, (4) K K∈Th → where R is the circumradius of K. Then the FEM converges in H1(Ω). We K notethatalltheseconditionsarederivedbytakingthepiecewiselinearLagrange interpolation Π u of u in Céa’s lemma. h Conditions (2), (3) are sufficient conditions for O(h) convergence. It was shown in [6] that the maximum angle condition is not necessary for (1) to hold. Theargument issimple andcan be directlyextended tothe circumradius condition. We take a system of triangulations satisfying (3) – thus exhibiting O(h) convergence – and refine each K arbitrarily to obtain ˜. Thus h h ˜ can contain an arbitrary amount of ar∈bitTrarily bad ‘degenerating’Telements, h Thowever since ˜ is a refinement of , it also exhibits O(h) convergence. h h T T Since (2)–(3) are only sufficient and not necessary, the question is what is a necessaryandsufficientconditionfor(1)tohold. Theonlystepinthisdirection is the Babuˇska-Aziz counterexample of [1] consisting of a regular triangulation of the square consisting only of ‘degenerating’ elements violating the maximum angle condition (except for several cut-off elements adjoining to the boundary ∂Ω), cf. Figure 8. Recently a more detailed, optimal analysis of this counterexample was performed in [11]. By controlling the speed of degeneration of 2 the aspect ration of the elements, one can produce a counterexample to O(h) convergenceandevenanexampleofnonconvergenceoftheFEM.Fromthisone counterexample,BabuˇskaandAzizconcludethatthemaximumanglecondition is essential for O(h) convergence, carefully avoiding the word necessary. Inthispaper,wewillbeconcernedwithamoregeneralversionof(1),namely O(hα) estimates of the form u U C(u)hα, (5) 1 | − | ≤ for α [0,1]. We will derive a necessary condition on the mesh geometry and ∈ also a new sufficient condition for (5) to hold. To the author’s best knowledge, the derived necessary condition is the first nontrivial necessary condition for O(h) or any other convergence of the finite element method, apart from the trivial necessary condition h 0. We note that the results of this paper are in → fact about the approximation properties of the piecewise linear finite element space with respect to the H1(Ω)-seminorm, rather than about the FEM itself. The structure of the paper is as follows. In Section 2, we derive a necessary condition for FEM convergence, i.e. (5) to hold. The condition roughly states that certain structures , bands of neighboring elements forming h B ⊂ T an alternating pattern such as in Figure 3 cannot be too long if the maximum angles of elements in go to π sufficiently fast w.r.t. h 0. The necessary B → condition is based on an estimate from below of the error u U on the H1( ) | − | B band –Theorem1forasinglebandandTheorem2for containingmultiple h B T bands. Corollaries 9 and 13 then state (very roughly, cf. Remark 12) that if (5) holds and if has length L Ch2α/5 then π α Ch3 2αL for all K . K − B ≥ − ≥ ∈ B Thisisthecaseofoneband , formultiplebandsthiscanbeimprovedto h B ⊂T π α Ch1 αL. Theseresultsenableustoconstructmanysimplecounterex- K − − ≥ amples to O(hα) convergence, which is done in Sections 2.2 and 2.4. We note thattheBabuˇska-Azizcounterexample(where consistsonlyoftheconsidered h T bands, cf. Figure 8) can be obtained by this technique. In fact, we recover the optimal results of [11]. However unlike the Babuˇska-Aziz counterexample, the regular periodic structure of is not necessary in the presented analysis. h T InSection3wedealwithsufficientconditionsfor(5)tohold. Forsimplicity we deal with the O(h) case, i.e. α = 1 and the general case is then a simple extension (Remark 18). We split into two parts: 1 consisting of elements Th Th K satisfying(3)forachosenα and 2 consistingofK violatingthiscondition 0 Th (‘degenerating’ elements). While on 1 it is safe to use Lagrange interpolation, Th on 2 we use a modified Lagrange interpolation operator to construct a linear Th function vK on K which is O(h )-close to u in the H1(K)-seminorm indepen- h K dently of the shape of K. The price paid is that vK no longer interpolates u in h A , the maximal-angle vertex of K, exactly but with a small perturbation of K theorderO(h2 ), cf. Lemma19. Thequestionthenariseshowtoconnectthese K piecewise linear functions continuously, if they no longer interpolate the continuous function u exactly in some vertices. Instead of changing the interpolation procedure, we change the interpolated function u itself, so that the Lagrange interpolationofthenewfunctionu˜ correspondstothemodifiedLagrangeinter- polation of u on 2. Then it is possible to prove that u Π u˜ Ch, hence Th | − h |1 ≤ 3 u U Ch by Céa’s lemma. For this purpose, we introduce the concept of 1 | − | ≤ correction functions and Sections 3.3, 3.4 are devoted to constructing the correction functions corresponding to different situations. Roughly speaking, h T can contain degenerating elements which can form arbitrarily large structures, chains,cf. Figure12,aslongastheirmaximal-angleverticesarenottoocloseto other vertices from 2. In general, if the degenerating elements form nontrivial Th clusters, such as the bands of Section 2, then these can have diameter up to O(hα/2). These results are contained in Theorems 5 and 6. Wenotethatwhileanecessaryand sufficientconditionforanykindofFEM convergence remains an open question, the gap between the derived conditions is small in some special cases, cf. Remark 20. For example, in the case of h T containingonebandoflengthLconsistingofsufficientlydegeneratingelements, the necessary condition for O(hα) convergence is L Ch2α/5 = Ch0.4α, while ≤ the sufficient condition is L Chα/2 =Ch0.5α. ≤ 1.1 Problem formulation and notation Let Ω R2 be a bounded polygonal domain with Lipschitz continuous boundary. W⊂e treat the following problem: Find u:Ω R2 R such that ⊂ → ∆u=f, u =u . (6) ∂Ω D − | ThereareseveralpossibilitieshowtotreatnonhomogeneousDirichletboundary conditions in the finite element method, here we consider the standard lifting technique, cf. [3]. We note that this choice is not necessary nor important in this paper, as we are essentially interested in the best H1-approximation of u in the discrete space, independent of the specific form of the weak formulation. Let g H1(Ω) be the Dirichlet lift, i.e. an arbitrary function such that ∈ g = u , and write u = u +g, u H1(Ω). Defining V = H1(Ω) and the |∂Ω D 0 0 ∈ 0 0 bilinearforma(u,v)= u vdx,thecorrespondingweakformof(6)reads: Ω∇ ·∇ Find u V such that 0 ∈ (cid:82) a(u+g,v)=(f,v), v V. ∀ ∈ The finite element method constructs a sequence of spaces X on conforming triangulations of Ω, where X H{1(Ωh)}hc∈o(n0s,his0t)s of {Th}h∈(0,h0) h ⊂ globally continuous piecewise linear functions on . Furthermore, let V = h h T X V. We do not assume any properties of , since it is the goal of this h h ∩ T papertoderivenecessaryandalsosufficientgeometricpropertiesof forfinite h T element convergence. We choose some approximation g X of g, e.g. a piecewise linear La- h h ∈ grange interpolation of g on ∂Ω such that g(x) = 0 in all interior vertices. We seek the FEM solution U X in the form U = U +g , where U V . The h 0 h 0 h ∈ ∈ FEM formulation then reads: Find U V such that 0 h ∈ a(U +g ,v )=(f,v ), v V . 0 h h h h h ∀ ∈ 4 v K˜1 K0 α˜ K1 Figure 1: Three neighboring elements K ,K˜ ,K . 0 1 1 Formally, weshouldusethenotationU insteadofU toindicatethatU X , h h h ∈ however we drop the subscript h for simplicity of notation. One must have this in mind e.g. when dealing with estimates of the form u U Ch. H1(Ω) | − | ≤ For simplicity of notation, Sobolev norms and seminorms on Ω will be denoted using simplified notation, e.g. := , := , etc. 1 H1(Ω) 2, W2,∞(Ω) |·| |·| |·| ∞ |·| Throughout the paper, C will denote a generic constant independent of h and othergeometricquantitiesof . TheconstantC hasingeneraldifferentvalues h T in different parts of the paper, even within the same chain of inequalities. 2 A necessary condition for FEM convergence Throughout Section 2 we will assume the following bound on the error. From this error estimate we will derive necessary conditions on the geometry of the triangulations for this estimate to hold. h T Error assumption: There exists α [0,1] independent of h such that for all ∈ u H2(Ω), h (0,h ) 0 ∈ ∈ u U C(u)hα. (7) 1 | − | ≤ Remark 1. Especially of interest are the cases α = 1 and α = 0 corresponding to O(h) convergence and (possibly) nonconvergence of the FEM. We note that we can just as well consider the more general case of u U C(u)f(h) H1(Ω) | − | ≤ for some (nonincreasing) function f. For simplicity we consider only (7) since this coincides well with the concept of orders of convergence. Remark 2. Assuming (7) holds for all u H2(Ω) is not necessary, however it ∈ is typical for FEM analysis. In Section 2, we will only use (7) for one specific quadratic function (19). Hence H2(Ω) can be replaced by any space that contains quadratic functions. Remark 3. WenotethatU in(7)neednotbetheFEMsolution,sincewedonot usetheFEMformulationinanywayinthefollowing. Weareessentiallydealing onlywithapproximationpropertiesofthespaceV : IfthereexistsU V such h h ∈ that (7) holds, what are the necessary properties of . h T The basic result upon which we will build the estimates of Section 2 is a simple geometric identity concerning gradients of a continuous piecewise linear function U (not necessarily the FEM solution) on a triplet of neighboring elements K ,K˜ ,K in the configuration from Figure 1. We denote U := U , U := U0 a1nd1U˜ := U . The key observation is that if U ,U 0are giv|Ken0, 1 |K1 1 |K˜1 0 1 5 ξ ∇U0 ∇U0⊥ A b v K˜1 K0 ∇U˜1 α˜ e K1 Bb π α˜ − Figure 2: Proof of Lemma 1. then U˜ is uniquely determined due to inter-element continuity. We note that 1 the following result is not an estimate, it is an equality, hence optimal. Lemma 1. Let ξ be the angle between the vectors (U U ) and v, where v 0 1 corresponds to the common edge of K ,K˜ , cf. Figu∇re 1.−Then 0 1 cos(ξ) (U˜ U ) = (U U ), (8) 1 1 0 1 |∇ − | sin(π α˜)|∇ − | − where α˜ is the maximum angle of K˜ . 1 Proof: By globally subtracting the function U (i.e. its extension to the whole 1 R2) from all functions on K ,K˜ ,K , we can assume that U 0 without loss 1 1 0 1 ofgenerality. FromthecontinuityofU,wehaveU (A)=U˜ (A≡),whereAisthe 0 1 common vertex of K and K˜ denoted in Figure 2. Furthermore, by continuity 0 1 and since U 0, then U˜ =0 on the edge e. Therefore 1 1 ≡ U˜ (A) (U˜ U ) = U˜ = | 1 | . (9) 1 1 1 |∇ − | |∇ | v sin(π α˜) | | − On the other hand, by continuity and since U 0, we have U (B) = 0 and 1 0 ≡ U = 0 on the line l passing through B in the direction U = (U U ) . 0 ∇ 0⊥ ∇ 0− 1 ⊥ Therefore, U˜ (A) 1 (U U ) = U = | | . (10) 0 1 0 |∇ − | |∇ | v cos(ξ) | | Expressing U˜ (A) from (10) and substituting into (9), we obtain the desired 1 result. | | (cid:3) Remark 4. Since (a,b)= a b cosα, where α is the angle between vectors a,b, | || | we can reformulate (8) as 1 (U˜ U ) = 1 v, (U U ) . (11) |∇ 1− 1 | sin(π α˜) v ∇ 0− 1 | | − (cid:12)(cid:0) (cid:1)(cid:12) (cid:12) (cid:12) In other words, on the right-hand side of (11), we have the magnitude of the projection of (U U ) into the direction given by the unit vector 1 v. ∇ 0− 1 v | | 6 Ki−1 K˜i Ki Γ Figure 3: A band of elements with edge Γ. B Lemma 1 states that if (U U ) is nonzero, then (U˜ U ) is huge, 0 1 1 1 |∇ − | |∇ − | since it is magnified by the factor 1 which tends to as α˜ π. If u sin(π α˜) ∞ → is e.g a reasonable quadratic function a−nd if U is a good approximation of 1 u then U 1, hence U˜ will be hug∇e, hence a bad approximation of ∇ |K1 |∇ 1|≈ ∇ 1 u , which is reasonably bounded. ∇ |K˜1 We will proceed as follows: We will choose an exact solution u with nonzero secondderivatives(aquadraticfunction). If u U issmall,thenwecanexpect 1 | − | (U U ) = 0, since this is essentially an approximation of second order 0 1 d|∇erivat−ives. T|h(cid:54) erefore (U˜ U ) will be enormous on degenerating triangles 1 1 due to the factor 1/sin|(∇π α˜−). Hen|ce U˜ cannot be a good approximation of 1 − ∇ u , therefore u U cannot be small. From this we can obtain restrictions |K˜1 | − |1 e.g. on sin(π α˜) for (7) to hold. The main difficulty is that the assumption of − u U being small does not imply (U U ) = 0 for any given triplet of 1 0 1 | − | |∇ − | (cid:54) elementssuchasinFigure1,weknowthisonly“onaverage”,intheL2(Ω)-sense duetotheH1(Ω)errorestimate. However,ifweconnecttheconsideredtriplets of elements into larger structures (bands , cf. Figure 3), then if these bands B are large enough, for some triplet of elements in necessarily (U U ) =0 0 1 B |∇ − |(cid:54) and we can apply Lemma 1. Our goal will be to prove that if (7) holds and is long enough, then the differences (U U ) must be nonzero on aveBrage, hence U˜ is a bad i+1 i i |∇ − | ∇ approximation of u and therefore the global error cannot be O(hα) on the ∇ |K˜i elements K˜ . Such contradictions allow us to formulate a necessary condition i for (7) and to construct counterexamples to O(h) and other convergences. Definition 2. We define a band as the union =( N K ) ( N K˜ ) such that K ,K˜ ,K for i =B1,⊂..T.,hN are neighboBring ∪elie=m0enits∪fo∪rmi=in1gia i 1 i i − triplet as in Lemma 1, such that their maximum angles form an alternating pattern as in Figure 3. We denote the union of longest edges of all K by Γ. i Remark 5. Inthefollowing, weassumethattheedgeΓofband liesonaline, B i.e. is ‘straight’. This makes the notation and proofs less technical, however B it is not a necessary assumption. We will indicate in the relevant places how the proofs should be modified to allow for ‘curved’. B Lemma 3. Let be a band as in definition 2 and let the error estimate (7) B hold. Then for sufficiently small h u U Chα, (12) L2(Γ) (cid:107) − (cid:107) ≤ where C depends only on Ω and the constant in (7). 7 Ω f Γ Ω Ω nΓ \ f Figure 4: Proof of Lemma 3 – splitting of Ω by Γ. Proof: The proof is essentially a straightforward application of the trace inequality. However, we must be careful, since in the standard trace inequality the constant depends heavily on the geometry of the domain. Therefore, we give a detailed proof of (12). We proceed similarly as in [5]. Forsimplicity,wedenotee:=u U. Wedefinetheconstantvectorfunction − µ= n , where n is the unit outer normal to on Γ, cf. Figure 4. Then µ is Γ Γ − B the unit outer normal to Ω on Γ, where Ω Ω is the subdomain containing ⊂ B defined by the line on which Γ lies. On one hand, we have (cid:101) (cid:101) (e2) µdx=2 e e µdx. ∇ · ∇ · (cid:90)Ω(cid:101) (cid:90)Ω(cid:101) On the other hand, by Green’s theorem, (e2) µdx= e2µ ndS, ∇ · · (cid:90)Ω(cid:101) (cid:90)∂Ω(cid:101) since divµ=0. As µ n=1 on Γ, by combining these two equalities we get · e2dS 2 e e µdx e2µ ndS, (13) ≤ ∇ · − · (cid:90)Γ (cid:90)Ω(cid:101) (cid:90)∂Ω(cid:101)∩∂Ω where we have omitted the boundary integral over ∂Ω (∂Ω Γ), which is \ ∪ nonnegativesinceµ n=1onthispartof∂Ω. Thesecondright-handsideterm · in (13) can be estimated as (cid:101) (cid:101) e2µ ndS e2dS Ch4, (14) · ≤ ≤ (cid:90)∂Ω(cid:101)∩∂Ω (cid:90)∂Ω since on ∂Ω, U is the piecewise linear Lagrange interpolation of u, hence e = | | u U Ch2 on ∂Ω. The first right-hand side term of (13) can be estimated | − | ≤ using Young’s inequality by 2 e e µdx e2+ e µ2dx e 2 + e2 . (15) ∇ · ≤ |∇ · | ≤(cid:107) (cid:107)L2(Ω) | |H1(Ω) (cid:90)Ω(cid:101) (cid:90)Ω Combining (13)–(15) with (7), we get e 2 e 2 +Ch4+Ch2α. (16) (cid:107) (cid:107)L2(Γ) ≤(cid:107) (cid:107)L2(Ω) 8 In (16), it remains to estimate e 2 by e2 . Ordinarily, one could (cid:107) (cid:107)L2(Ω) | |H1(Ω) simplyusePoincaré’sinequality,howevere / H1(Ω). Nonetheless,asmentioned ∈ 0 above,e isoftheorderO(h2),whichallowsustoobtainasimilarestimateto ∂Ω | Poincaré’s inequality. Using Green’s theorem and the trivial identity ∂x1 = 1, ∂x1 we estimate ∂e e 2 = 1e2dx= x n e2dS x 2e dx. (17) (cid:107) (cid:107)L2(Ω) 1 1 − 1 ∂x (cid:90)Ω (cid:90)∂Ω (cid:90)Ω 1 Without loss of generality, let 0 Ω. Therefore we can estimate (17) using ∈ Young’s inequality as e 2 diamΩ e 2 +2diamΩ e e (cid:107) (cid:107)L2(Ω) ≤ (cid:107) (cid:107)L2(Γ) (cid:107) (cid:107)L2(Ω)| |H1(Ω) 1 diamΩ e 2 + e 2 +(diamΩ)2 e2 . ≤ (cid:107) (cid:107)L2(Γ) 2(cid:107) (cid:107)L2(Ω) | |H1(Ω) Therefore, e 2 2diamΩ e 2 +2(diamΩ)2 e2 Ch4+Ch2α. (18) (cid:107) (cid:107)L2(Ω) ≤ (cid:107) (cid:107)L2(Γ) | |H1(Ω) ≤ Combining (16) and (18) gives us estimate (12) for sufficiently small h. (cid:3) Remark 6. If we do not assume that Γ lies on a straight line, cf. Remark 5, we need to find a constant vector function µ such that µ n µ for some µ >0. 0 0 · ≥ This can be done e.g. by taking µ = nΓ for some i 0,...,N , i.e. the Ki ∈ { } normal to K on Γ, and assuming that nΓ nΓ µ > 0 for all j = i, i.e. i Ki · Kj ≥ 0 (cid:54) B does not ‘bend too much’. The constant µ then figures in the left-hand sides 0 of (13) and (16). Consequently, we get the constant 1/µ in the right-hand side 0 of (12). In the following, we shall assume that the exact solution of (6) is u(x ,x )=x2+x2, (19) 1 2 1 2 i.e. f = 4withcorrespondingDirichletboundaryconditions. IfListhelength − of Γ, then Γ can be parametrized by t (0,L) as ∈ Γ= x=a+tg,t (0,L) , { ∈ } where a is an endpoint of Γ and g = 1 is a vector in the direction of Γ. The | | restriction u can then also be parametrized by t (0,L). We denote the Γ | ∈ corresponding function u and observe that Γ u (t)=u(a+tg)=t2+p (t), (20) Γ 1 wherep P1(0,L),i.e. alinearfunction. Wehavethefollowingapproximation 1 ∈ result: Lemma 4. Let u P2(0,L) be given by (20). Let Π1 be the L2(0,L)- Γ ∈ (0,L) orthogonal projection onto P1(0,L). Then u Π1 u = 1 L5/2. (21) (cid:107) Γ− (0,L) Γ(cid:107)L2(0,L) 6√5 9 Proof: For convenience, u can be rewritten in the form u (t) = (t L/2)2+ Γ Γ − p˜ (t), wherep˜ P1(0,L). Thenu Π1 u =(t L/2)2 Π1 (t L/2)2, 1 1 ∈ Γ− (0,L) Γ − − (0,L) − since Π1 p˜ =p˜ . (0,L) 1 1 On (0,L), the quadratic function (t L/2)2 is symmetric with respect to − L/2,henceitsprojectionmustbeaconstantfunction. Itcanbethereforeeasily computed that Π1 (t L/2)2 = 1 L3. Therefore the norm in (21) can be (0,L) − 12 straightforwardly computed, giving the desired result. To save space, we omit the elementary, yet lengthy calculations. (cid:3) Corollary 5. Let α [0,1] and L Ch2α/5. Then ∈ ≥ u Π1 u Chα. (22) (cid:107) Γ− (0,L) Γ(cid:107)L2(0,L) ≥ Remark 7. Specifically of interest are the cases α = 1, corresponding to L ≥ Ch2/5 and α=1, corresponding to L C. These two cases will lead to neces- ≥ sary conditions and counterexamples to O(h) convergence, and convergence of the FEM, respectively. Remark 8. If Γ does not lie on a straight line, cf. Remark 5, the length- parametrized restriction u : (0,L) R of u will be continuous, piecewise Γ Γ → | quadratic. However, if Γ is close to a straight line, then u will be close to the Γ globally quadratic function t2+p (t) of (20). We can then estimate 1 u Π1 u (cid:107) Γ− (0,L) Γ(cid:107)L2(0,L) (23) t2+p (t) Π1 u u t2+p (t) . ≥ (cid:107) 1 − (0,L) Γ(cid:107)L2(0,L)−(cid:107) Γ− 1 (cid:107)L2(0,L) (cid:12) (cid:12) For the first(cid:12)right-hand side term we have (cid:12) t2+p (t) Π1 u t2+p (t) Π1 (t2+p (t)) = 1 L5/2 (cid:107) 1 − (0,L) Γ(cid:107)L2(0,L) ≥(cid:107) 1 − (0,L) 1 (cid:107)L2(0,L) 6√5 by Lemma 4. If we assume e.g. that u t2+p (t) 1 L5/2 (24) (cid:107) Γ− 1 (cid:107)L2(0,L) ≤ 12√5 then by (23) we get the statement of Lemma 4 as an estimate from below with the lower bound 1 L5/2. Condition (24) is effectively a condition on how Γ 12√5 deviates from a straight line. NowwefocusonthediscretesolutiononΓ. Again,ifΓliesonastraightline, then the length-parametrized restriction U := U L2(0,L) is a continuous Γ Γ | ∈ piecewiselinearfunctiononthepartitionof(0,L),i.e. Γ,inducedby . Hence h T U H1(0,L). Γ ∈ For convenience, to deal with functions from H1(0,L) instead of H1(0,L), 0 we define U¯ P1(0,L) such that U¯(0)=U (0) and U¯(L)=U (L). Let Γ Γ ∈ w :=U U¯, (25) Γ − then w(0)=w(L)=0. We have the following estimate. 10