Recently, we devised an approach to a posteriori error analysis, which
clarifies the role of oscillation and where oscillation is bounded in terms of the current
approximation error. Basing upon this approach, we derive plain convergence
of adaptive linear finite elements approximating the Poisson problem. The result
covers arbritray H^-1-data and characterizes convergent marking strategies

Kreuzer, Christian

Veeser, Andreas

Eldorado - Ressourcen aus und für Lehre, Studium und Forschung

                No. 583            March 2018   Convergence of adaptive finite element  methods with error-dominated oscillation  C. Kreuzer, A. Veeser                    ISSN: 2190-1767 Convergence of adaptive finite elementmethods with error-dominated oscillationChristian Kreuzer1 and Andreas Veeser21 Technische Universita¨t Dortmund, Fakulta¨t fu¨r Mathematik, Vogelpothsweg 87,44227 Dortmund christian.kreuzer@tu-dortmund.de2 Universita` degli Studi di Milano, Dipartimento di Matematica F. Enriques, ViaC. Saldini 50, 20133 Milano andreas.veeser@unimi.itAbstract. Recently, we devised an approach to a posteriori error analysis, whichclarifies the role of oscillation and where oscillation is bounded in terms of the cur-rent approximation error. Basing upon this approach, we derive plain convergenceof adaptive linear finite elements approximating the Poisson problem. The resultcovers arbritray H−1-data and characterizes convergent marking strategies.1 IntroductionAdaptive finite element methods (AFEMs) are well-established and efficienttools for solving boundary value problems. A common form is the iterationofSOLVE → ESTIMATE → MARK → REFINE, (1)i.e. solve for the current finite element solution, estimate its error by means ofso-called a posteriori indicators, and use this information in a marking strat-egy to make refinement decisions. The resultant problem-dependent adapta-tion of the mesh often significantly improves efficiency, compared to classicaluniform refinement. There are theoretical results corroborating this practi-cal observation; see the surveys [3,9] and [5], which even obtains a nonlinearquasi-optimality result with respect to an augmented energy norm error.Almost all results about the convergence behavior of AFEMs invoke extraregularity for the data which does not fit to the proven convergence rate in thelight of [1,6]. To exemplify and explain this, let us consider the approximationof the weak solution of the Poisson problem−∆u = f in Ω ⊂ Rd, u = 0 on ∂Ω (2)by a realization of (1) with linear finite elements. Here one usually requires theregularity f ∈ L2(Ω) beyond H−1(Ω), irrespective of the actual regularity ofthe target function u. One reason for requiring f ∈ L2(Ω) lies in a posteriorierror analysis. Aiming at computability, local H−1-norms are replaced withL2-norms scaled by the local meshsize hT , leading to terms like the classicaloscillation indicator hT minfT∈R ‖f − fT ‖L2(T ), which needs f ∈ L2(Ω) to bedefined. Note however that such a term is not yet really computable and may2 C. Kreuzer and A. Veesercause serious overestimation; cf. [7]. Another reason lies in many convergenceanalyses themselves. They often rely on the fact that the scaling factors ofthe aforementioned L2-norms strictly reduce under refinement and, e.g., inthe case of oscillation terms, this strict reduction hinges on extra regularityof f . It is worth mentioning that the second reason does not apply to thederivations of plain convergence in [8,10].In [7] we present a new approach to a posteriori analysis covering arbitraryf ∈ H−1(Ω) and prove in particular‖∇(u− UT )‖Ω . ET + δT . ‖∇(u− UT )‖Ω ,where UT is the linear finite element solution over the mesh T , the estimatorET is computable in a similar sense as UT , while the computation of theoscillation δT requires additional information on f .Here we prove plain convergence of adaptive linear finite elements for (2)by combining the a posteriori analysis [7] and the convergence analyses [8,10].The result forgoes extra regularity and so convergence rates are precluded.Section 2 presents the adaptive finite method in detail, along with an accountof [7], and section 3 proves its convergence, characterizing convergent markingstrategies. Section 4 then concludes with a brief discussion of a few convergentmarking strategies.2 Adaptive algorithm with abstract marking strategyIn this section we introduce adaptive finite element methods for approximat-ing the weak solution u ∈ H10 (Ω) of (2) for some fixed but arbitrary loadf ∈ H−1(Ω). (3)Here H10 (Ω) is the subspace of Sobolev functions in H1(Ω) with vanishingtrace and H−1(Ω) denotes its topological dual space.Bisection and conforming linear finite elements. Let T0 be a suitable sim-plicial, face-to-face (conforming) mesh of Ω and denote by T the family ofits refinements using iterative or recursive bisection; cf. the discussion in [9]and the references therein. In what follows, ‘.’ stands for ‘≤ C’, where thegeneric constant C may depend on the shape coefficient of T0. For example,for any simplex arising in T, we have that its shape coefficient . 1.Let T ∈ T be a mesh. We denote the set of its vertices by VT and theset of its faces by FT . The associated space of continuous piecewise affinefunctions isV(T ) := {V ∈ C0(Ω¯) | V|T ∈ P1 for all T ∈ T } ⊂ H1(Ω).Its nodal basis {ΦT,z}z∈VT is given byΦT,z ∈ V(T ) such that ΦT,z(y) = δzy for all z, y ∈ VT .AFEM convergence with error-dominated oscillation 3Note suppΦT,z =⋃{T ∈ T : T 3 z} =: ωT,z. The finite element functionsV˚(T ) := {V ∈ V(T ) | ∀z ∈ VT ∩ ∂Ω V (z) = 0} ⊂ H10 (Ω) are conformingand the associated Galerkin approximation UT of (2) is characterized byUT ∈ V˚(T ) such that ∀V ∈ V˚(T )∫Ω∇UT · ∇V dx = 〈f, V 〉, (4)where 〈·, ·〉 is the duality pairing between H−1(Ω) and H10 (Ω). In order tobe able to compute UT , we suppose that∀T ∈ T, z ∈ VT ∩Ω 〈f, ΦT,z〉 is available. (5)A posteriori error analysis with error-dominated oscillation. We outline theapproach to a posteriori error analysis in [7], which prepares the ground forformulating our convergence theorem. To this end, we fix the mesh T anddrop it as subscript in the following discussion. Further, given a subdomainω ⊂ Ω, we choose ‖∇ · ‖ω := ‖∇ · ‖L2(ω) as norm on H10 (ω) and denote by‖ · ‖−1;ω its dual norm on H−1(ω).We then have the global relationship ‖∇(u − U)‖Ω = ‖f + ∆U‖−1;Ωbetween error and residual as well as, locally,∀z ∈ V ‖f +∆U‖−1;ωz ≤ ‖∇(u− U)‖ωz . (6)Exploiting Galerkin orthogonality and the fact that {Φz}z∈V provides a par-tition of unity, one derives that, for any v ∈ H10 (Ω) and V ∈ V˚(T ),〈f +∆U, v〉 = 〈f +∆U, v − V 〉 .∑z∈V‖f +∆U‖−1,ωz‖∇v‖ωz ; (7)see, e.g., [4,7]. Applying the Cauchy-Schwarz inequality on the sum and re-calling (6), one obtains the error localization‖∇(u− U)‖2Ω h∑z∈V‖f +∆U‖2−1;ωz .In general, upper bounds for the local residual norms ‖f +∆U‖−1;ωz , asfor the global one ‖f +∆u‖−1;Ω , cannot be computed by a finite number ofoperations from the information in (5). The reason lies in the possibly infinite-dimensional nature of f ; see [7]. One therefore may isolate this difficultyby inserting a finite-dimensional approximation of f , e.g., some piecewiseconstant function f¯ . Unfortunately, the arising terms ‖f − f¯‖−1;ωz , or thelarger classical oscillation terms ‖h(f − f¯)‖ωz , may overestimate the error byfar on a given mesh, see [7], and even lead to worse convergence rates, see [4].To overcome these drawbacks, we have developed in [7] a projection oper-ator P : H−1(Ω) → D(T ) with the following properties. The evaluation Pfis computable under (5) and the rangeD(T ) := {` ∈ H−1(Ω) | 〈`, v〉 = ∑T∈T∫TcT v dx+∑F∈F∫FcF v dsfor all v ∈ H10 (Ω) with cT , cF ∈ R for T ∈ T , F ∈ F}4 C. Kreuzer and A. Veesercontains ∆U ∈ H−1(Ω). Moreover, P is locally stable and invariant in that,for any ` ∈ H−1(Ω), z ∈ V, and D ∈ D(T ), we have‖P`‖−1;ωz . ‖`‖−1;ωz , (8a)as well as` = P` in H−1(ωz) whenever ` = D in H−1(ωz). (8b)Consequently, we have P(∆U) = ∆U and the local splittings‖f +∆U‖−1;ωz h ‖Pf +∆U‖−1;ωz + ‖f − Pf‖−1;ωz , (9)where ‖Pf + ∆U‖−1;ωz is quantifiable under (5), but the computation orbounding of ‖f − Pf‖−1;ωz requires information beyond. To sum up, weformulate the following theorem.Theorem 1 (A posteriori bounds). For any mesh T ∈ T, we have‖∇(u− UT )‖2Ω .∑z∈VT‖PT f +∆UT ‖2−1;ωT,z + ‖f − PT f‖2−1;ωT,zas well as, for any z ∈ VT ,‖PT f +∆UT ‖2−1;ωT,z + ‖f − PT f‖2−1;ωT,z . ‖∇(u− UT )‖2ωT,z .Adaptive algorithm. The bounds in Theorem 1 are vertex-wise, while bisec-tion is applied element-wise. Therefore, we assume that there are elementindicators ET (T ), T ∈ T , such that‖PT f +∆UT ‖2−1;ωT,z .∑T⊂ωT,zET (T )2 (10a)andET (T )2 .∑z∈T∩VT‖PT f +∆UT ‖2−1;ωT,z . (10b)Such local equivalences hold for various indicators resembling the ones fromthe standard residual estimator, the hierarchical estimator, or estimatorsbased upon discrete local problems; cf. [7].In order to achieve a similar grouping for the oscillation, we introduce,for any T ∈ T , its minimal ring ω(T ) := ⋂{ωT ′(T ) : T ′ ∈ T, T ′ 3 T}, whereωT (T ) := ∪{T ′ ∈ T : T ′ ∩ T 6= ∅}. Thenδ(T ) := ‖f − PT f‖−1;ω(T ) (11)does not depend on the surrounding mesh, and we can derive the inequalities‖f − PT f‖2−1;ωT,z .∑T⊂ωT,zδ(T )2, δ(T )2 .∑z∈T∩VT‖f − PT f‖2−1;ωT,z . (12)AFEM convergence with error-dominated oscillation 5Algorithm (AFEM). Starting from T0 and k = 0, compute (Vk)k, (Uk)k,(Tk)k iteratively as follows, where step 3 will be specified further below.1. Set Vk := V˚(Tk) and let Uk ∈ Vk be the solution of (4) with T = Tk.2. Compute the indicators of estimator and oscillation in (10) and (11),respectively, and write Ek as an abbreviation for ETk .3. Choose a subset Mk ⊂ Tk with the help of the values of the indicators{Ek(T )}T∈Tk and {δ(T )}T∈Tk .4. Let Tk+1 be the smallest refinement of Tk in T such thatMk ∩Tk+1 = ∅.5. Increment k and go to step 1.3 Convergence and marking strategyThe above adaptive method generalizes global uniform refinement, which isthe special case corresponding to Mk = Tk for all k. The new key featureis that certain elements may not be refined anymore. In other words: it mayhappen thatT ∗ := {T | ∃m ∈ N0 ∀k ≥ m T ∈ Tk} 6= ∅ and so V∞ :=⋃kVk 6= H10 (Ω).It is then the task of step 3, the marking strategy, to preclude u ∈ H10 (Ω)\V∞,which is equivalent to non-convergence.Let us first derive a necessary condition for the marking strategy. Iflimk→∞ ‖∇(u − Uk)‖Ω = 0, then, for any element T ∈ T ∗, the local lowerbounds in (10), (11), and Theorem 1 imply limk→∞ Ek(T ) + δ(T ) = 0 for theassociated indicators. Hence it is necessary for convergence that the markingstrategy ensures∀T ∈ T ∗ δ(T ) = 0 and limk→∞Ek(T ) = 0, (13)which specifies [8, (5.1)]. It turns out that this condition is also sufficientfor convergence, as it complements the following property of our refinementprocedure. Let hk denote the meshsize function of Tk given by hk|T = |T |1/dfor all T ∈ Tk and let χk the characteristic function of⋃{T ∈ Tk : T 6∈ Tk+1}.Then [9, Lemma 9] showslimk→∞‖hkχk‖L∞(Ω) = 0, (14)which generalizes maxΩ hk → 0 for non-adaptive, uniform refinement.Theorem 2 (Adaptive convergence). Let f ∈ H−1(Ω) be arbitrary andassume that the indicators of the estimator satisfy (10). Then the approximatesolutions (Uk)k of the AFEM converge to the exact solution u if and only ifthe marking strategy ensures (13).6 C. Kreuzer and A. VeeserProof. We adopt the proof of [9, Theorem 8], which essentially follows [10].Using the Lax-Milgram theorem, let U∞ ∈ V∞ be such that∫Ω∇U∞ · ∇V dx = 〈f, V 〉 for all V ∈ V∞.Thanks to [9, Lemma 7], we havelimk→∞‖∇(U∞ − Uk)‖Ω = 0 (15)and it remains to show that U∞ = u or, as an alternative, that its residualvanishes:0 = 〈R∞, v〉 := 〈f, v〉 −∫Ω∇U∞ · ∇v dx for all v ∈ H10 (Ω). (16)Here we can take the test functions only from C∞0 (Ω), because C∞0 (Ω) is adense subset of H10 (Ω). Consequently, the convergence (15) shows that (16)follows from∀ϕ ∈ C∞0 (Ω) limk→∞〈Rk, ϕ〉 = 0, (17)where Rk := f +∆Uk ∈ H−1(Ω) is the residual of Uk.In order to verify this, let ϕ ∈ C∞0 (Ω) and k, ` ∈ N0 with k ≥ ` andintroduce the set T ∗` := T` ∩ T ∗. The inclusion V` ⊂ Vk, the abstract errorbound (7), the local equivalences (9), as well as the upper bounds in (10) and(11) imply〈Rk, ϕ〉 = 〈Rk, ϕ− I`ϕ〉 . S`,k + S∗`,k, (18)where I` denotes Lagrange interpolation onto V` and we expect thatS`,k :=∑T∈Tk\T ∗`(Ek(T ) + δ(T ))‖∇(ϕ− I`ϕ)‖ωk(T )gets small because of decreasing meshsize, whereasS∗`,k :=∑T∈T ∗`(Ek(T ) + δ(T ))‖∇(ϕ− I`ϕ)‖ωk(T )gets small thanks to condition (13) on the marking strategy. Here ωk(T ) =ωTk(T ) is the ring around T in Tk.We first deal with S`,k. The lower bounds in (10), (11), and Theorem 1entail that∑T∈Tk\T ∗`(Ek(T ) + δ(T ))2 . ‖∇(Uk − U∞)‖2Ω + ‖∇(U∞ − u)‖2Ω . 1 (19)is uniformly bounded thanks to (15). Furthermore, standard error bounds forLagrange interpolation on T` yield∑T∈Tk\T ∗`‖∇(ϕ− I`ϕ)‖2ωk(T ) .∑T∈T`\T ∗`‖∇(ϕ− I`ϕ)‖2ω`(T ). ‖h`χ`‖2L∞(Ω)‖D2ϕ‖2L∞(Ω)|Ω|.AFEM convergence with error-dominated oscillation 7Hence the Cauchy-Schwarz inequality on the sum S`,k and (14) implyS`,k → 0 as `→∞ uniformly in k. (20)For S∗`,k, condition (13) leads to∑T∈T ∗`(Ek(T ) + δ(T ))2 = ∑T∈T ∗`Ek(T )2 → 0 as k →∞and, since∑T∈T ∗` ‖∇(ϕ− I`ϕ)‖2L2(ωk(T ))is uniformly bounded, the Cauchy-Schwarz inequality providesS∗`,k → 0 as k →∞ for any fixed `. (21)To conclude, let  > 0 be arbitrary. We exploit (20) and (21) by firstchoosing ` so that S`,k ≤ /2 and next k ≥ ` so that S∗`,k ≤ /2. Insertingthis into (18) yields the desired convergence (17) and finishes the proof. uunionsq4 Convergent marking strategiesIn this concluding section, we discuss how to ensure condition (13) on themarking strategy, which is necessary and sufficient for the convergence of theAFEM.To this end, the following observation about a vanishing limit is helpful.If (Tk)k∈N0 is a sequence of elements satisfying Tk ∈ Tk \ Tk+1, k ∈ N0, thenEk(Tk) + δ(Tk) . ‖∇(u− Uk)‖ωk(Tk)≤ ‖∇(U∞ − Uk)‖ωk(Tk) + ‖∇(u− U∞)‖ωk(Tk) → 0(22)as k →∞. In fact, the first term on the right-hand side vanishes thanks to (15)and the second term vanishes in view of ωk(Tk) . |Tk| . ‖hkχk‖dL∞(Ω) andproperty (14).Condition (13) is thus satisfied if we havemaxT ′∈Tk(δ(T ′) + Ek(T ′)) ≤ maxT ′∈Tk\Tk+1(δ(T ′) + Ek(T ′)),which, thanks toMk ⊂ Tk \ Tk+1, can be achieved by requiring that at leastone element with maximal combined indicators is marked:Mk ∩{T ∈ Tk | δ(T ) + Ek(T ) ≥ maxT ′∈Tk(δ(T ′) + Ek(T ′))} 6= ∅. (23)This property is verified by most of the common marking strategies appliedto the combined indicators δ(T ) + Ek(T ), T ∈ Tk. Important examples aremaximum-, Do¨rfler-, and equal distribution strategy; cf. [8].8 C. Kreuzer and A. VeeserAs an alternative to combined marking, one may mark the two indicatorsseparately. Similarly as before, requiring (23) for the single indicator Ek(T )instead of the combined indicator Ek(T ) + δ(T ) then implies∀T ∈ T ∗ limk→∞Ek(T ) = 0.To ensure also δ(T ) = 0 for all T ∈ T ∗, one may employ again the respectivecounterpart of (23) or a different approach, which, as fast tree approxima-tion [2], capitalizes on the locality and history of the oscillation indicator.The following simple consequence of Theorem 2, which is also of interest byits own, is useful in this context.Remark 3 (Modified oscillation). For any simplex T in T, let δ˜(T ) be a mod-ification or approximation of δ(T ) such that δ˜(T ) = 0 =⇒ δ(T ) = 0. Thenthe AFEM, where each δ(T ) is replaced by δ˜(T ), converges if and only iflimk→∞ Ek(T ) = 0 and δ˜(T ) = 0 for all T ∈ T ∗.Acknowledgment. AV gratefully acknowledges the support of the GNCS,which is a part of the Italian INdAM.References1. P. Binev, W. Dahmen, R. DeVore, and P. Petrushev, Approximationclasses for adaptive methods, Serdica Math. J. 28:4 (2002), 391–416, Dedicatedto the memory of Vassil Popov on the occasion of his 60th birthday.2. P. Binev and R. DeVore, Fast computation in adaptive tree approximation,Numer. Math. 97:2 (2004), 193–217.3. C. Carstensen, M. Feischl, and D. Praetorius, Axioms of adaptivity,Comput. Math. Appl. 67:6 (2014), 1195–1253.4. A. Cohen, R. DeVore, and R. H. Nochetto, Convergence rates of AFEMwith H−1 data, Found. Comput. Math. 12:5 (2012), 671–718.5. L. Diening, C. Kreuzer, and R. Stevenson, Instance optimality of theadaptive maximum strategy, Found. Comput. Math. 16:1 (2016), 33–68.6. F. D. Gaspoz and P. Morin, Approximation classes for adaptive higher orderfinite element approximation, Math. Comp. 83:289 (2014), 2127–2160.7. C. Kreuzer and A. Veeser, Oscillation in a posteriori error estimation, inpreparation.8. P. Morin, K. G. Siebert, and A. Veeser, A basic convergence result forconforming adaptive finite elements, Math. Models Methods Appl. Sci. 18:5(2008), 707–737.9. R. H. Nochetto and A. Veeser, Primer of adaptive finite element methods,Multiscale and adaptivity: modeling, numerics and applications, Lecture Notesin Math., vol. 2040, Springer, Heidelberg, 2012, pp. 125–225.10. K. G. Siebert, A convergence proof for adaptive finite elements without lowerbound, IMA J. Numer. Anal. 31:3 (2011), 947–970.

Convergence of adaptive finite element methods with error-dominated oscillation

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Convergence of adaptive finite element methods with error-dominated oscillation

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