Guaranteed and robust a posteriori bounds for Laplace eigenvalues and eigenvectors: a unified framework

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1 Guaranteed and robust a posteriori bounds for Laplace eigenvalues and eigenvectors: a unified framework Eric Cancès, Geneviève Dusson, Yvon Maday, Benjamin Stamm, Martin Voralík To cite tis version: Eric Cancès, Geneviève Dusson, Yvon Maday, Benjamin Stamm, Martin Voralík. Guaranteed and robust a posteriori bounds for Laplace eigenvalues and eigenvectors: a unified framework. Numerisce Matematik, Springer Verlag, 08, <0.007/s >. <al v3> HAL Id: al ttps://al.inria.fr/al v3 Submitted on Jul 08 HAL is a multi-disciplinary open access arcive for te deposit and dissemination of scientific researc documents, weter tey are publised or not. Te documents may come from teacing and researc institutions in France or abroad, or from public or private researc centers. L arcive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recerce, publiés ou non, émanant des établissements d enseignement et de recerce français ou étrangers, des laboratoires publics ou privés.

2 Guaranteed and robust a posteriori bounds for Laplace eigenvalues and eigenvectors: a unified framework Eric Cancès Geneviève Dusson Yvon Maday Benjamin Stamm Martin Voralík July, 08 Abstract Tis paper develops a general framework for a posteriori error estimates in numerical approximations of te Laplace eigenvalue problem, applicable to all standard numerical metods. Guaranteed and computable upper and lower bounds on an arbitrary simple eigenvalue are given, as well as on te energy error in te approximation of te associated eigenvector. Te bounds are valid under te sole condition tat te approximate i-t eigenvalue lies between te exact (i )-t and (i+)-t eigenvalue, were te relative gaps are sufficiently large. We give a practical way ow to ceck tis; te accuracy of te resulting estimates depends on tese relative gaps. Our bounds feature no unknown (solution-, regularity-, or polynomial-degree-dependent) constant, are optimally convergent (efficient), and polynomial-degree robust. Under a furter explicit, a posteriori, minimal resolution condition, te multiplicative constant in our estimates can be reduced by a fixed factor; moreover, wen an elliptic regularity assumption on te corresponding source problem is satisfied wit known constants, tis multiplicative constant can be brougt to te optimal value of wit mes refinement. Applications of our framework to nonconforming, discontinuous Galerkin, and mixed finite element approximations of arbitrary polynomial degree are provided, along wit numerical illustrations. Our key ingredient are equivalences between te i-t eigenvalue error, te associated eigenvector energy error, and te dual norm of te residual. We extend tem in an appendix to te generic class of bounded-below self-adjoint operators wit compact resolvent. Key words: Laplace eigenvalue problem, a posteriori estimate, guaranteed eigenvalue bound, guaranteed eigenvector error bound, abstract framework, nonconforming finite element metod, discontinuous Galerkin metod, mixed finite element metod Introduction Precise numerical approximation of eigenvalues and eigenvectors of elliptic operators on general domains is crucial in countless applications. In addition to standard conforming Galerkin (variational) approximations, Université Paris-Est, CERMICS, Ecole des Ponts, 6 & 8 Av. Pascal, Marne-la-Vallée, France. (cances@cermics.enpc.fr). Inria, rue Simone Iff, Paris, France (martin.voralik@inria.fr). Matematics Institute, University of Warwick, Coventry CV47AL, United Kingdom (g.dusson@warwick.ac.uk). Sorbonne Université, Université Paris-Diderot SPC, CNRS, Laboratoire Jacques-Louis Lions, LJLL, F Paris (maday@ann.jussieu.fr). Institut Universitaire de France, 75005, Paris, France. Division of Applied Matematics, Brown University, 8 George St, Providence, RI 09, USA. Center for Computational Engineering Science, RWTH Aacen University, Aacen, Germany (stamm@matcces.rwt-aacen.de). Computational Biomedicine, Institute for Advanced Simulation IAS-5 and Institute of Neuroscience and Medicine INM-9, Forscungszentrum Jülic, Germany. Tis work was supported by te ANR project MANIF Matematical and numerical issues in first-principle molecular simulation. Part of tis work as been supported from Frenc state funds managed by te CalSimLab LABEX and te ANR witin te Investissements davenir program (reference ANR--LABX ). Te last autor as also received funding from te European Researc Council (ERC) under te European Union s Horizon 00 researc and innovation program (grant agreement No GATIPOR).

3 nonconforming metods suc as nonconforming finite elements, discontinuous Galerkin elements, or mixed finite elements are very popular, and one naturally asks te question of te size of te error in teir eigenvalue and eigenvector approximations. Te issue of error control is usually tackled via te a posteriori estimates teory. Recently, powerful estimates were obtained for nonconforming approximations of te Laplace source problem, see Destuynder and Métivet [9], Ainswort [, ], Kim [48, 49], Voralík [6], Carstensen and Merdon [], or Ern et al. [35, 37, 38], see also te references terein. Te Laplace eigenvalue problem seems to be structurally more difficult. Recently, toug, guaranteed a posteriori estimates on te error in te i-t eigenvalue ave been obtained in Carstensen and Gedicke [0] and Liu [5]. Te teory of [0, 5] applies for arbitrarily coarse meses and gives convincing numerical results in many test cases. One could, owever, comment tat tese results only seem to apply to te lowest-order nonconforming finite element metod, te arguments used are of a priori nature tat relies on an interpolation estimate, and an overestimation in presence of singularities and graded meses may appear as te bounds feature te diameter of te largest mes element, see [0, Section 6.3]. More precisely, tese bounds loose accuracy if te diameter of te largest mes element does not tend to zero. Armentano and Durán [6], Luo et al. [53], Hu et al. [45, 46], or Yang et al. [63] also derived (guaranteed) eigenvalue estimates for te nonconforming finite element metod, were, owever, a saturation assumption may be necessary and/or te results are valid only asymptotically. Errors in bot eigenvalue and eigenvector approximations in nonconforming metods ave also been studied previously, altoug rater seldom. We refer in particular to Dari et al. [6] for nonconforming finite elements, to Giani and Hall [40] for discontinuous Galerkin elements, and to Durán et al. [34] and Jia et al. [47] for mixed finite elements. Unfortunately, tese estimates systematically contain solution-independent but unknown constants as well as solution-dependent, uncomputable terms, claimed iger order on fine enoug meses via a priori arguments. Te purpose of te present paper is to extend our conforming teory of Cancès et al. [7] to a general framework for guaranteed and optimally convergent a posteriori bounds for bot arbitrary simple i-t eigenvalue and te associated eigenvector of te Laplace eigenvalue problem. Let Ω R d, d =, 3, be a polygonal/polyedral domain wit a Lipscitz boundary. Let te exact eigenvector and eigenvalue pairs (u i, λ i ) satisfy u i = λ i u i in Ω, (.) wit te condition u i =, see (.) below for te precise weak formulation. We denote by (, ) te L (Ω) or [L (Ω)] d scalar product over Ω and by te associated norm and let H (T ) := {v L (Ω); v K H (K) K T } (.) be te so-called broken Sobolev space, were te traces on mes faces do not need to coincide. It is defined over a computational mes T of te domain Ω; details of te setting are given in Section. On H (T ), we generalize te usual weak gradient of H (Ω) to te discrete gradient θ featuring a parameter θ {, 0, }, see (.5) below. We consider ere an abstract setting were te approximate eigenvector eigenvalue pair (u i, λ i ) is not necessarily linked to any particular numerical metod, u i H (T ) is a piecewise polynomial possibly nonconforming in te sense tat u i H0 (Ω) and not necessarily scaled to u i =, λ i R +, and te relation θ u i = λ i typically does not old at te discrete level in contrast to te continuous one. Concrete examples of numerical metods fitting to our setting can be found later in Section 7. Our main tools are an equilibrated flux reconstruction σ i H(div, Ω) satisfying σ i = λ i u i and an eigenvector reconstruction s i H0 (Ω), bot defined in Section 3. Tese are piecewise polynomials suc tat σ i and s i are as close as possible to te discrete gradient θ u i. Tey are constructed over patces of mes elements following Destuynder and Métivet [30], Braess and Scöberl [3], Carstensen and Merdon [], and Ern and Voralík [38], see also te references terein. We employ tem in Section 4 to sow in particular ow te dual norm of te residual of te pair (s i, λ i ) can be bounded in a computable way. Section 5 ten bounds te L (Ω)-norm of te Riesz representation of te residual of (s i, λ i ) under an assumption of elliptic regularity on te corresponding Laplace source problem. It enables later to give improved computable estimates in te considered nonconforming setting. Our main results are collected in Section 6 and crucially rely on [7, Section 3], were mutual relations between te i-t eigenvalue error, te associated eigenvector energy error, and te dual norm of te residual in terms of an arbitrary pair ( s i, λ i ) H0 (Ω) R + suc tat s i = are given. Using s i := s i / s i,

4 were s i is te above eigenvector reconstruction, inequality (6.4) of Teorem 6. in particular gives s i η i λ i, (.3a) were η i is an a posteriori error estimator wit a typical structure η i = m i (λ i u i s i / λ + s i + σ i )/ s i. (.3b) Tus (.3a) gives a guaranteed and computable lower bound for te i-t exact eigenvalue λ i. An upper bound on λ i is recalled in inequality (6.4) of Teorem 6.3. A guaranteed and computable a posteriori estimate on te associated eigenvector energy error is given next, see in particular estimate (6.6) of Teorem 6.4 revealing θ (u i u i ) η i + θ (u i s i ). (.3c) Te eigenvalue and eigenvector error bounds (.3) are efficient (optimally convergent) in te sense tat η i + θ (u i s i ) C i ( θ (u i u i ) + consistency terms + norm of mean values of jumps of u i ), (.4) see inequality (6.7) of Teorem 6.5. Here C i is a generic constant tat only depends on λ, λ i, on te lower bound λ i+ of λ i+, possibly on te upper bound λ i of λ i, on te sape of Ω, and on some (broken) Poincaré Friedrics constants over patces of elements and a stability constant of mixed finite elements (bot only depending on te sape regularity of te mes and on te space dimension d). Te constant C i is in particular independent of te polynomial degree of u i, leading to polynomial-degree robustness. Te consistency terms above may not be present, typically for nonconforming finite elements. Similarly, te jump mean values of u i are zero in nonconforming and mixed finite elements, and tis term also vanises in our developments for te symmetric variant of te discontinuous Galerkin finite element metod wen te parameter θ equals. Te above results are valid under te condition (6.), requesting λ i to lie between computable bounds λ i+ and λ i (wen i > ) on te surrounding exact eigenvalues λ i+ and λ i, see te discussion in [7, Remark 5.4] for its practical verification. We also need te residual ortogonality condition of Assumption 3. in order to reconstruct te equilibrated flux. Ten, for u i a piecewise polynomial of degree p, te reconstructions s i and σ i are prescribed in discrete spaces of order p +. Tere is no specific condition on te fineness of te mes in Case A of Teorems 6. and 6.4, but te multiplicative factor m i in (.3b) contains te relative gap of te form max{( λ i λ i ), ( λ i λ i+ ) }. Two improvements are possible. Under te computable minimal resolution criterion (6.6b) (satisfied for fine enoug meses), m i is reduced by a fixed factor, see Case B of Teorems 6. and 6.4. If an elliptic regularity assumption on te corresponding source problem is satisfied and if te minimal resolution condition (6.8b) olds, te relative gap in m i is multiplied by a power of te mes size, so tat m i can be brougt to te optimal value of in te limit of mes size tending to zero, at least for nonconforming, mixed, and symmetric discontinuous Galerkin metods, wic we sow in Case C of Teorems 6. and 6.4. Te efficiency constant C i from (.4) or (6.7) can be fully traced from te detailed estimates of Teorem 6.5; in particular it deteriorates for increasing eigenvalues. Te application of our abstract results to a given numerical metod merely requires te verification of te setting and of Assumption 3.. We undertake tis in Section 7 for nonconforming finite elements, discontinuous Galerkin elements, and mixed finite elements of arbitrary polynomial degree. For mixed finite elements, elementwise postprocessings of te approximate eigenvector and of its flux need to be performed first. Numerical experiments are presented in Section 8 for te nonconforming and discontinuous Galerkin metods and fully support our teoretical findings for a couple of model problems. In particular, te a posteriori applicability conditions (6.6b) and (6.8b) for cases B and C are satisfied ere already on very coarse meses. Some concluding remarks and an outlook are given in Section 9. In particular, inexact algebraic eigenvalue solvers promoted in Mermann and Miedlar [55] or Carstensen and Gedicke [0] can be treated as in [7] and te references terein, allowing to generalize te present estimates to an arbitrary iterative solver step were Assumption 3. typically does not old. We finis our paper by two extensions. We first sow in Appendix A tat te key relations between te i-t eigenvalue error, te associated eigenvector energy error, and te dual norm of te residual, wen u i is conforming, i.e., u i H 0 (Ω), are in fact not restricted to te Laplace operator; we extend tem to 3

5 te generic class of bounded-below self-adjoint operators wit compact resolvent in Teorems A. and A.. Appendix B ten gives a furter possible improvement of te first eigenvalue upper bound: from λ s of (6.5) in Teorem 6.3 to λ s η of Proposition B.3. We only treat ere simple eigenvalues and associated eigenvectors; clustered and multiple eigenvalues are dealt wit in contribution [8] tat we are now finalizing. We rely tere on te framework of density matrices and actually provide guaranteed bounds for an arbitrary set of eigenvalues under te sole condition tat tere is a sufficient gap between tis set and te surrounding eigenvalues. Tis analysis is done in te abstract framework of Appendix A. Setting Let H (Ω) be te Sobolev space of L (Ω) functions wit weak gradients in [L (Ω)] d. We denote encefort by V := H 0 (Ω) its zero-trace subspace. Later, we will also employ te space H(div, Ω) of [L (Ω)] d functions wit weak divergences in L (Ω).. Te Laplace eigenvalue problem Te weak formulation of (.) looks for (u i, λ i ) V R + wit u i = and ( u i, v) = λ i (u i, v) v V. (.) It is well-known (see, e.g., Babuška and Osborn [8] or Boffi [0] and te references terein) tat u i, i, form a countable ortonormal basis of L (Ω) consisting of eigenvectors from V ; we assume tat te sequence of eigenvalues is suc tat 0 < λ < λ... λ i. We will actually suppose tat te eigenvalue λ i tat we study is simple, wic is always te case for i =. Te associated eigenvector u i is ten uniquely defined upon fixing its sign by te condition (u i, χ i ) > 0, were χ i L (Ω) is typically a caracteristic function of Ω (for i = ) or of its subdomain (for i > ). Te setting for i = in particular implies te Poincaré Friedrics inequality v λ v v V. (.). Meses and generic piecewise polynomial spaces We denote by T a matcing simplicial mes in te sense of Ciarlet [4], sape-regular wit a parameter κ T > 0: te ratio of te element diameter K and of te diameter of te inscribed ball to K T is bounded by κ T (uniformly for a sequence of meses). Denote also by te maximal element diameter over all K T. Te mes (d )-dimensional faces are collected in te set E, wit interior faces E int and boundary faces E ext. A generic face is denoted by e, its diameter by e, and its unit normal vector (te direction is arbitrary but fixed) by n e. We will often employ te jump operator [ ] yielding te difference of te traces of te argument from te two mes elements tat sare e E int along n e and te actual trace for e E ext. Similarly, te average operator { } yields te mean value of te traces from adjacent mes elements on interior faces and te actual trace on boundary faces. Te set of vertices will be denoted by V ; it is composed of interior vertices V int and vertices located on te boundary V ext, wit a generic vertex denoted by a. Let P q (K) stand for polynomials of total degree at most q 0 on K T, and P q (T ) H (T ) for piecewise q-t order polynomials on T. For a given q, we denote by V q := P q(t ) V te q-t order conforming finite element space. Similarly, for q 0, V q := {v H(div, Ω); v K [P q (K)] d + P q (K)x} and Q q := P q(t ) stand for te Raviart Tomas Nédélec mixed finite element spaces of order q, cf. Brezzi and Fortin [5] or Roberts and Tomas [59]. We will also use te lowest-order broken space V 0 (T ) := {v [L (Ω)] d ; v K [P 0 (K)] d + P 0 (K)x}, were in contrast to V q, no normal trace continuity is imposed via te inclusion in H(div, Ω)..3 Broken and discrete gradients On te broken Sobolev space H (T ) defined in (.), te usual weak gradient is not defined. We will in tis paper use two successive generalizations of te notion of te weak gradient. We will first denote by 4

6 v [L (Ω)] d te broken gradient of v H (T ) given by ( v) K := (v K ) K T. (.3) We will need te following generalization of (.), te so-called broken Poincaré Friedrics inequality, see Brenner [4, Remark.] or Voralík [6, Teorem 5.4] and te references terein: v C bf v + e E int e Π 0 e [v ] e + v, Ω v H (T ), (.4) were Π 0 e stands for te L (e)-ortogonal projection onto constants on te face e,, denotes te L (Ω) scalar product over Ω, and te constant C bf only depends on te domain Ω, te space dimension d, and te mes sape regularity parameter κ T. In order to prove te elliptic regularity bound of Proposition 5.4 below in a very general setting, we are lead to a furter generalization. It is motivated by te lifting operators used in te discontinuous Galerkin finite element metod, see Di Pietro and Ern [3, Section 4.3] and te references terein, but we crucially rely ere on te space V 0 (T ) of te lowest-order broken Raviart Tomas Nédélec polynomials defined above. Let v H (T ). For eac face e E, we define te lifting operator l e : L (e) V 0 (T e ), were T e regroups te mes elements saring te face e and V 0 (T e ) is te restriction of V 0 (T ) tereon. Te lifting l e ([v ]) is prescribed by (l e ([v ]), v ) Te = {v } n e, [v ] e for all v V 0 (T e ). We ten extend l e ([v ]) by zero outside of T e to form an element of V 0 (T ). For a parameter θ {, 0, }, te discrete gradient θ v [L (Ω)] d is ten given by θ v := v θ e E l e ([v ]). (.5) We observe tat θ v = v wen θ = 0 or wen te jumps of v are of mean value 0, i.e., [v ], e = 0 for all e E ; indeed, tis follows from te fact tat v n e are constants for v V 0 (T ). Bot broken and discrete gradients are consistent extensions of te weak gradient in te sense tat.4 Residual and its dual norm θ v = v = v v H 0 (Ω). (.6) Te derivation of a posteriori error estimates usually exploits te concept of te residual and of its dual norm. We will proceed in tis way as well. Trougout te paper, it will turn out to be convenient to employ te residual of different pairs (w i, λ i ) H (T ) R, were we take for w i te approximate solution u i, te eigenvector reconstruction s i of Definition 3.3 below, or a generic function in V. Let V stand for te dual of V. Definition. (Residual and its dual norm). For any pair (w i, λ i ) H (T ) R, define te residual Res θ (w i, λ i ) V by Res θ (w i, λ i ), v V,V := λ i (w i, v) ( θ w i, v) v V. (.7a) Its dual norm is ten Res θ (w i, λ i ) := sup Res θ (w i, λ i ), v V,V. (.7b) v V, v = We will also often work wit te Riesz representation of te residual r wi V, given by ( r wi, v) = Res θ (w i, λ i ), v V,V v V. (.8a) Ten r wi = Res θ (w i, λ i ). (.8b) 5

7 3 Eigenvector and equilibrated flux reconstructions We introduce in tis section two key reconstructions, following [58, 50, 9,, 48, 49, 6, 35,, 3, 37, 38] and te references terein. To motivate, note tat from (.), it is straigtforward tat u i H(div, Ω), wit te weak divergence equal to λ i u i. On te discrete level, owever, θ u i H(div, Ω) in general, and, a fortiori, ( θ u i ) λ i u i. We will tus introduce an equilibrated flux reconstruction, a vector-valued function σ i constructed from te given pair (u i, λ i ), satisfying σ i H(div, Ω), σ i = λ i u i. (3.a) (3.b) Similarly, as we treat ere cases were u i V, possibly jumping between te mes elements, we will employ an eigenvector reconstruction, a scalar-valued function s i constructed from u i and satisfying s i V. (3.) Actually, bot σ i and s i will be piecewise polynomials defined in standard finite element subspaces of H(div, Ω) and V, respectively. 3. Ortogonality of te residual Let ψ a for a V stand for te piecewise affine function taking value at te vertex a and zero at te oter vertices. Remarkably, tese functions form a partition of unity via a V ψ a = Ω. Denote by T a te patc of elements of T wic sare te vertex a V and by ω a te corresponding subdomain of Ω. Recall tat θ {, 0, } is te fixed parameter from te definition of te discrete gradient (.5). Our key assumption will be: Assumption 3. (Ortogonality of te residual to at functions). Tere olds λ i (u i, ψ a ) ωa ( θ u i, ψ a ) ωa = Res θ (u i, λ i ), ψ a V,V = 0 a V int. Assumption 3. can typically be verified for an exact algebraic solver; Section 7 below sows ow to ceck it for some standard numerical metods. Inexact solvers, were Assumption 3. does not old, can be treated as in [7] and te references terein. 3. Reconstruction spaces In practice, te approximate eigenvector u i is a piecewise polynomial, u i P p (T ) for some p. To define te reconstructions in tis setting, we will, for eac vertex a V, work wit restrictions V p+ (ω a ) and Q p+ (ω a ) of te spaces from Section. to te patc subdomain ω a ; conversely, we will often tacitly extend functions defined on ω a by zero outside of ω a. Wit n ωa standing for te outward unit normal of ω a, we define V a := {v V p+ (ω a ); v n ωa = 0 on ω a }, Q a := {q Q p+ (ω a ); (q, ) ωa = 0}, a V int, V a := {v V p+ (ω a ); v n ωa = 0 on ω a \ Ω}, Q a := Qp+ (ω a ), a V ext, W a := P p+ (T a ) H0 (ω a ) a V. 3.3 Equilibrated flux reconstruction We construct σ i satisfying (3.) by local constrained minimizations: Definition 3. (Equilibrated flux reconstruction). Let (u i, λ i ) P p (T ) R be arbitrary but satisfying Assumption 3.. Prescribe σi a Va by solving σi a := arg min ψ a θ u i + v v V a, v ωa a V (3.3a) =Π Q a (ψ aλ i u i θ u i ψ a) 6

8 and define σ i := a V σ a i. (3.3b) In (3.3a), Π Q a stands for te L (Ω)-ortogonal projection onto te local space Q a. It is actually only needed for te simplification of Remark 6.3 below; oterwise, ψ a λ i u i θ u i ψ a is a piecewise polynomial of degree p + on te patc T a, wit mean value zero tanks to Assumption 3., so tat Π Q a (ψ a λ i u i θ u i ψ a ) = ψ a λ i u i θ u i ψ a. Imposing te divergence of σi a in tis way and defining σ i via (3.3b) leads to (3.b), see, e.g., [38, Lemma 3.5]. It is easy to verify tat problems (3.3a) are equivalent (see [38, Remark 3.7]) to te mixed finite element approximation to te omogeneous Neumann (Neumann Diriclet close to te boundary) problem posed on te patc T a : find σi a Va and pa Qa suc tat (σ a i, v ) ωa (p a, v ) ωa = (ψ a θ u i, v ) ωa v V a, ( σ a i, q ) ωa = (ψ a λ i u i θ u i ψ a, q ) ωa q Q a. It follows from te standard references [5, 59] tat wit te considered coice of te spaces V a, Qa, te discrete inf sup condition is satisfied. 3.4 Eigenvector reconstruction For nonconforming eigenvectors u i, i.e., u i is a piecewise polynomial not included in V = H 0 (Ω) but merely in H (T ), te eigenvector reconstruction complying wit requirement (3.) is obtained via local unconstrained minimizations employing te broken gradient (.3): Definition 3.3 (Eigenvector reconstruction). Let u i P p (T ) be arbitrary. Prescribe s a i W a by solving s a i := arg min (ψ a u i v ) ωa a V (3.4) v W a and define s i := a V s a i. Problems (3.4) are equivalently described by teir Euler Lagrange conditions; tese request to find te conforming finite element approximation s a i W a to te omogeneous Diriclet problem posed over te patc T a suc tat ( s a i, v ) ωa = ( (ψ a u i ), v ) ωa v W a. 4 Dual norm of te residual and nonconformity bounds We summarize ere bounds on te dual norm of te residual and on nonconformity tat are available from te context of source problems. Tey will be crucial later in Section Some additional notation and useful inequalities We first need to introduce some more background. Consider a vertex a V and on te patc domain ω a define H (ω a ) := {v H (ω a ); (v, ) ωa = 0}, a V int, (4.a) H (ω a ) := {v H (ω a ); v = 0 on ω a Ω}, Ten te Poincaré Friedrics inequality, corresponding to (.) on te patces ω a, states a V ext. (4.b) v ωa C PF,ωa ωa v ωa v H (ω a ), (4.a) 7

9 were C PF,ωa depends only on te mes regularity parameter κ T and te space dimension d. Similarly, wen te functions are piecewise H only, we will use te inequality v ωa C bpf,ωa ωa v ω a + e E, a e e Π 0 e [v ] e (4.b) valid for all v H (T ) suc tat (v, ) ωa = 0 wen a V int, and were te constant C bpf,ω a depends only on κ T and d. Inequality (4.b) may be seen as a local version of (.4) on te patc domain ω a, wit te mean value condition (v, ) ωa = 0 for a V int or appearance of boundary faces e E ω a for a V ext. Tis replaces te boundary term v, Ω from (.4). Define C cont,pf := max a V { + C PF,ωa ωa ψ a,ωa } and C cont,bpf := max a V { + C bpf,ωa ωa ψ a,ωa }. Te constants C cont,pf and C cont,bpf only depend on te mes regularity parameter κ T and te space dimension d and can be fully estimated from above, see te discussion in [38, proofs of Lemmas 3. and 3.3 and Section 4.3.]. In particular, tere olds, see Carstensen and Funken [9, Teorem 3.] or Braess et al. [, Section 3] (ψ a v) ωa C cont,pf v ωa v H (ω a ), a V. (4.3) 4. Stability of te equilibrated flux and eigenvector reconstructions Recently, Costabel and McIntos [5, Corollary 3.4], Demkowicz et al. [8, Teorem 7.], and Demkowicz et al. [7, Teorem 6.] ave sown fundamental results on te rigt inverse of respectively te divergence, te normal trace, and te trace operators for polynomial data on a single (reference) tetraedron. Terefrom, te two following key stability results for te constructions of Definitions 3. and 3.3 follow: ψ a θ u i + σ a i ωa C st sup v H (ωa); v ωa = Res θ (u i, λ i ), ψ a v V,V, (ψ a u i s a i) ωa C st inf v H 0 (ωa) (ψ a u i v) ωa, (4.4a) (4.4b) were te constant C st > 0 only depends on te mes sape regularity parameter κ T and te space dimension d. Indeed, (4.4a) as been sown in Braess et al. [, Teorem 7] in two space dimensions and Ern and Voralík [39, Corollaries 3.3 and 3.6] in tree space dimensions, wereas (4.4b) is proven in [38, Corollary 3.6] in two space dimensions and [39, Corollary 3.] in tree space dimensions. In [39, Corollaries 3.3 and 3.6], we merely need to set τ p = ψ a θ u i, r K = ψ a (λ i u i + ( θ u i )) K for any simplex K in te patc T a, and r F = ψ a [ θ u i ] n F for any face F in te patc T a to infer (4.4a) for interior vertices. Similarly, to see (4.4b), it is enoug to take τ p = ψ a u i and r F = ψ a [u i ] F in te notation of [39, Corollary 3.]. We also remark tat computable upper bounds on C st are discussed in [38, Lemma 3.3]. 4.3 Dual norm of te residual and nonconformity bounds Our a posteriori error estimates and teir efficiency below will rely on te two following intermediate results: Corollary 4. (Upper and lower bounds on te dual norm of te residual). Let (u i, λ i ) P p (T ) R + satisfy Assumption 3. and let σ i, s i be respectively constructed following Definitions 3. and 3.3. Ten ( ) λi Res θ (s i, λ i ) u i s i + s i + σ i, (4.5a) λ θ u i + σ i (d + )C st C cont,pf Res θ (u i, λ i ). (4.5b) Proof. Let v V wit v = be fixed. Using te definition of te residual (.7a), te consistency of te definition of te discrete gradient (.6), adding and subtracting (σ i, v), and employing te Green teorem and te equilibrium property (3.b), Res θ (s i, λ i ), v V,V = λ i (s i, v) ( s i, v) = (λ i s i σ i, v) ( s i + σ i, v) = λ i (s i u i, v) ( s i + σ i, v). 8

10 Tus, te caracterization (.7b) of te dual norm of te residual, te Caucy Scwarz inequality, and te Poincaré Friedrics inequality (.) yield (4.5a). To sow (4.5b), let us first note tat sup v H (ωa); v ωa = Res θ (u i, λ i ), ψ a v V,V Res θ (u i, λ i ),ωa Res θ (u i, λ i ),ωa C cont,pf, sup v H (ωa); v ωa = (ψ a v) ωa (4.6) were Res θ (u i, λ i ),ωa := sup v H 0 (ω a); v ωa = Res θ (u i, λ i ), v V,V, using tat for any v H (ω a ), ψ a v H0 (ω a ) and (4.3). Since ( θ u i + σ i ) K = a V K (ψ a θ u i + σi a ) K for any K T, were V K stands for te set of te vertices of te element K, and since any simplex as d + vertices, θ u i + σ i = (ψ a θ u i + σ a i) K K T a V K (d + ) ψ a θ u i + σi a K K T a V K = (d + ) ψ a θ u i + σi a ω a. a V Now relying on (4.4a) and (4.6), we infer θ u i + σ i (d + )CstC cont,pf Res θ (u i, λ i ),ω a. a V Finally, an estimate for combination of negative norms on recovering subdomains, see, for example, [3, Teorem 3.] and [9, Teorem 3.5] and te references terein, implies (4.5b). Corollary 4. (Nonconformity lower bound). For (u i, λ i ) P p (T ) R +, let s i be constructed following Definition 3.3. Ten ( (u i s i ) (d + ) C stc cont,bpf (u i u i ) + d(d + )C stc cont,bpf e e E Π 0 e [u i ] e ). (4.7) Proof. Tis result can be sown as in [38, Lemma 3. and Section 4.3.], relying on (4.b) and crucially on (4.4b). 5 Elliptic regularity bounds on te Riesz representation of te residual An important ingredient for our estimates is a bound on r si of te Riesz representation r si V of te residual Res θ (s i, λ i ) given by (.8a). We now derive a sarp estimate on r si under an elliptic regularity assumption. Let ζ (r si ) be te weak solution of te Laplace source problem ζ (r si ) = r si in Ω and ζ (r si ) = 0 on Ω, i.e., ζ (r si ) V is suc tat ( ζ (r si ), v) = (r si, v) v V. (5.) We will use an Aubin Nitsce duality argument, see te references in [7, Section 5.] for te conforming setting and, e.g., Antonietti et al. [3, 4] and te references terein for te nonconforming setting. Recalling te lowest-order H0 (Ω)- and H(div, Ω)-conforming finite element spaces V and V0 from Section., and denoting Π 0 te L (Ω)-ortogonal projection onto piecewise constants, let: 9

11 Assumption 5. (Elliptic regularity). Te solution ζ (r si ) of problem (5.) belongs to te space H +δ (Ω), 0 < δ, so tat te approximation and stability estimates min (ζ v V (r si ) v ) C I δ ζ (r si ) H +δ (Ω), (5.a) ζ (r si ) H +δ (Ω) C S r si (5.b) are satisfied. Let moreover, for a suitable v V 0 suc tat v = Π 0 (r si ), te approximation and stability estimates ζ (r si ) + v C I δ ζ (r si ) H +δ (Ω), v C S ζ (r si ) (5.3a) (5.3b) old. Let finally te inverse inequality v n e e C inv e v K K T, e E K (5.4) old for all v V 0, were E K stands for te faces of te simplex K. Remark 5. (Constants C I and C S ). Let Ω be a convex polygon in two space dimensions. Ten it is, so tat δ = and C S =, see Grisvard [4, Teorem ]. Ten, calculable bounds on C I can be found in Liu and Kikuci [5] and Carstensen et al. [], see also te references terein. In particular, on unstructured triangular meses, according to [5, equation (46)], + cos(θ K ) ν+ (α K, θ K ) K C I = max, K T sin(θ K ) K classical tat ζ (r si ) H (Ω) and ζ (r si ) H (Ω) = ζ (r si ) = r si were K is te medium edge lengt of K T, α K K is te minimum edge lengt of K, θ K is te angle between tem, and ν + (α K, θ K ) = + αk + + αk cos θ K + αk 4, see notation from Section, Figure, and equations (8), (36), and (46) in [5]. For a mes formed by isosceles rigt-angled triangles, C I from [5]. Remark 5.3 (Constants C I, CS, and C inv ). As above, te ideal case is ζ (r si ) H (Ω), wic appens in particular wen Ω is a convex polygon in two space dimensions. Ten δ = and calculable bounds on C I can be found in Mao and Si [54] and Carstensen et al. [] for te coice v as te Raviart Tomas Nédélec interpolate of ζ (r si ). In particular, following [], C I = max K T max α angle of K /4 + /j, cos(α), (5.5) were j, is te first positive root of te Bessel function J. Tis in particular gives C I = /4 + /j, 0.65 for a structured mes wit isosceles rigt-angled triangles. For tis interpolate, (5.3b) olds, witout any regularity assumption beyond ζ (r si ) L q (Ω), q >. Finally, (5.4) olds for any v V 0 and C inv only depends on te sape regularity of te mes and on te space dimension d, as v is from te lowest-order space. Proposition 5.4 (Elliptic regularity bound on r si ). Let (u i, λ i ) H (T ) R + and let Assumptions 3. and 5. old. Ten r si λ i u i s i + C I C S δ Res θ (u i, λ i ) + λ C I C S δ θ (u i s i ) { } e Π 0 e [u i ] e. e E + Π 0 (u i s i ) + θ d + C inv C S λ (5.6) 0

12 Proof. By te definition (5.) of ζ (r si ), te definition (.8a) of r si, te definition (.7a) of Res θ (s i, λ i ), and te ortogonality Assumption 3., r si = ( ζ (r si ), r si ) = λ i (s i, ζ (r si )) ( s i, ζ (r si )) were ζ V = λ i (s i u i, ζ (r si )) + λ i (u i, ζ (r si )) ( θ u i, ζ (r si )) ( θ (s i u i ), ζ (r si )) = λ i (s i u i, ζ (r si )) + λ i (u i, ζ (r si ) ζ ) ( θ u i, (ζ (r si ) ζ )) ( θ (s i u i ), ζ (r si )), is arbitrary. One more application of (.7a), (.8a) ten leads to r si = λ i (s i u i, ζ (r si )) + ( r ui, (ζ (r si ) ζ )) ( θ (s i u i ), ζ (r si )) λ i s i u i ζ (r si ) + r ui (ζ (r si ) ζ ) ( θ (s i u i ), ζ (r si )), were we ave also employed te Caucy Scwarz inequality. Now te Poincaré Friedrics inequality (.) / λ. For te second term above, we need to take te best ζ and employ te estimates (5.) to arrive at ( ) r si λi u i s i + C I C S δ λ r ui r si gives ζ (r si ) ζ (r si ) / λ and we ave from (5.) tat ζ (r si ) r si ( ζ (r si ), θ (s i u i )). Let now v V 0 be suc tat v = Π 0 (r si ). Definition (.5) of te discrete gradient and te fact tat v V 0 V0 (T ) give (v, θ u i ) = (v, u i ) + θ e E (v, l e ([u i ])) = (v, u i ) + θ e E {v } n e, [u i ] e. Tus, using tat v H(div, Ω) (so tat {v } n e = v n e ), s i V, and elementwise te Green teorem gives (v, θ (s i u i )) = ( v, s i u i ) K + (θ ) v n e, [u i ] e. K T e E Te last term above actually disappears wen te jumps of u i are of mean value 0, i.e., [u i ], e = 0 for all e E, or wen θ =. As v n e P 0 (e), we can, in general, at least replace [u i ] by Π 0 e [u i ] and estimate tis term using te inverse inequality (5.4) and Caucy Scwarz one, as eac simplex as d + faces v n e, Π 0 e [u i ] e { v K T ; e E K C inv e Π 0 } e [u i ] e e E e E { } d + C inv v e Π 0 e [u i ] e. e E

13 Tus, for v satisfying (5.3) and under te stability assumption (5.b), we infer ( ζ (r si ), θ (s i u i )) = ( ζ (r si ) + v, θ (s i u i )) + (Π 0 (r si ), u i s i ) + (θ ) v n e, Π 0 e [u i ] e e E = ( ζ (r si ) + v, θ (s i u i )) + (r si, Π 0 (u i s i )) + (θ ) e E v n e, Π 0 e [u i ] e C I C S r δ si θ (u i s i ) + r si r si + θ d + C inv C S λ { Π 0 (u i s i ) e Π 0 e [u i ] e e E }. Combining te above estimates wit te caracterization (.8b) of r ui finises te proof. 6 Guaranteed and computable upper and lower bounds in a unified framework We combine ere te results of Sections 4 and 5 togeter wit te key generic equivalences of [7, Section 3] to derive te actual guaranteed and computable eigenvalue and eigenvector bounds in a unified framework. 6. Eigenvalues We first tackle te question of upper and lower bounds for te i-t eigenvalue λ i. We refer to [7, Remark 5.4 and 5.5] for te discussion on obtaining te auxiliary eigenvalue bounds λ, λ i, λ i, and λ i+. Teorem 6. (Guaranteed lower bounds for te i-t eigenvalue). Let te i-t exact eigenvalue λ i, i, be simple and suppose te auxiliary bounds λ λ, λ i λ i, λ i+ λ i+, as well as λ i λ i wen i >, for λ, λ i, λ i+, λ i > 0. Let te approximate eigenvector eigenvalue pair (u i, λ i ) P p (T ) R + satisfy Assumption 3., as well as te inclusion λ i < λ i wen i >, λ i < λ i+. (6.) Let te equilibrated flux reconstruction σ i be given by Definition 3. and te eigenvector reconstruction s i by Definition 3.3, wit s i 0. Denote te principal estimator ( ) η i,res := λ i u i s i + s i + σ i (6.) s i λ togeter wit te discrete relative gaps { ( ) ( λi c i := max, λ ) } i, (6.3a) λ i λ i+ { ( ) ( c i := max λ λi i, λ i+ λ ) } i (6.3b) λ i λ i+ and te scaled eigenvector reconstruction Ten, te i-t eigenvalue lower bound is s i := s i s i. s i η i λ i, (6.4)

14 were te estimator η i takes different forms in te following tree cases: Case A (No smallness assumption) If te sign caracterization (u i, s i ) 0 is known to old, te lower i-t eigenvalue estimate (6.4) is valid wit η i := ( + (λ i + λ i ) c i)η i,res. (6.5) If only ( s i, χ i ) > 0 olds for te sign caracterization function χ i of Section., te factor in (6.5) needs to be replaced by ( s i Π i s i ), were Π i s i stands for te L (Ω)-ortogonal projection of s i on te span of χ i. Case B (Improved estimates under a smallness assumption) Assume te sign caracterization ( s i, χ i ) > 0 and define α i := c i η i,res, (6.6a) were α i is a computable bound on te L (Ω) error u i s i. Let te smallness assumption { (λ ) } α i min, χi ( s i, χ i ) λ i (6.6b) old, so tat in particular λi α i λ 4 is bounded by and tends to zero; wen i =, taking λ = λ i = λ i is possible and makes te fraction λ vanis. Ten te lower i-t eigenvalue estimate (6.4) olds wit λ i η i := c i ( λ i α ) i η λ 4 i,res. (6.7) Case C (Optimal estimates under elliptic regularity assumption) Assume te elliptic regularity of Assumption 5. togeter wit te sign caracterization ( s i, χ i ) > 0. Define te L (Ω) estimators α i of u i s i by ( ci λ i α i := u i s i + C I C S δ θ u i + σ i s i λ + C I C S δ θ (u i s i ) + Π 0 (u i s i ) { e Π 0 e [u i ] e e E + θ d + C inv C S λ } ). (6.8a) Ten, under te smallness assumption α i χ i ( s i, χ i ), (6.8b) te lower i-t eigenvalue estimate (6.4) olds wit η i given by η i := η i,res + (λ i + λ i )α i. (6.9) Remark 6. (Form of te complete estimator η i ). In Cases A and B above, we immediately see η i = m i η i,res, m i := { ( + (λi + λ i ) c i ) in Case A, ) c i ( λi α i λ 4 in Case B, (up to te possible replacement of te factor in Case A) (m i is bounded by c i and tends to ci in Case B). Tus te complete estimator η i indeed takes te form (.3b) announced in te introduction, were in particular te key role of η i,res given by (6.) and te unfavorable multiplication by te discrete relative gaps c i or c i of (6.3) are obvious. In Case C, ηi rater takes an additive form, wit te key estimator ηi,res. Note tat η i,res as a leading term (te second one) tat comes wit constant / s i tat tends to te optimal value of one. Te oter term in η i,res is typically of iger order since it is in te L (Ω)-norm (see also Remark 6.9 below). Te estimator ηi,res is supplemented by te term (λ i + λ i )α i. Tis last term 3

15 contains in α i a multiplication by te discrete relative gap c i. It is, owever, typically of iger order wenever te last contribution in α i of (6.8a) disappears, eiter wen te discrete gradient parameter θ from (.5) is taken as (e.g., symmetric discontinuous Galerkin finite elements), or wen te jumps are of mean value zero (e.g., mixed and nonconforming finite elements). Tis is in particular linked to te terms u i s i in te L (Ω) norm, see Remark 6.9 below, and to te presence of te δ-power of te maximal element diameter. Note in tis respect tat C I C S δ θ u i + σ i and C I C S δ θ (u i s i ) are still efficient estimates even if = const (some mes elements are not refined), see Teorem 6.5 below. Tus, rougly speaking, sall some mes elements stay unrefined, Case C looses optimal sarpness (te leading term is not only η i,res wit te optimal constant one) and Case C sort of degenerates to Case B. We now prove te tree cases of Teorem 6. separately: Proof of Teorem 6., Case A. It is immediate from estimate (4.5a) of Corollary 4. and from definition (6.) of te principal estimator η i,res togeter wit te scaling s i = s i / s i tat te dual norm of te residual of ( s i, λ i ) can be estimated as Res θ ( s i, λ i ) η i,res. (6.0) If (u i, s i ) 0 is known to old, define α i by (6.6a). From [7, Lemma 3.] in combination wit (.8b), we ten infer te L (Ω) bound u i s i c i Res θ ( s i, λ i ) α i. (6.) Now te upper bound in [7, Teorem 3.4], in combination wit te first bound of [7, Teorem 3.5], gives s i λ i (u i s i ) Res θ ( s i, λ i ) + (λ i + λ i )α i, (6.) and one more use of (6.0) proves (6.4) wit η i given by (6.5). If only ( s i, χ i ) > 0 olds, we take α i := ( s i Π i s i ) ci η i,res and proceed as in [7, proof of Teorem 5., Case A] to find u i s i ( s i Π i s i ) ci Res θ ( s i, λ i ) α i instead of (6.), and we conclude as above. Proof of Teorem 6., Case B. Te second condition in (6.6b) implies tat assumptions of [7, Lemma 3.3] are satisfied for s i. Tus te L (Ω) bound (6.) is valid for α i given by (6.6a). Te first condition in (6.6b) ten allows us to use te improved estimate in [7, Teorem 3.5]. In combination wit te upper bound in [7, Teorem 3.4], tis gives and we conclude by (6.0). s i λ i (u i s i ) c i ( λ i α ) i Res θ ( s i, λ i ) λ 4, (6.3) Proof of Teorem 6., Case C. Proposition 5.4 togeter wit te bound Res θ (u i, λ i ) θ u i + σ i and te scaling s i = s i / s i imply r si α i ci, were α i is given by (6.8a). Next, condition (6.8b) implies tat assumptions of [7, Lemma 3.3] are satisfied for s i and consequently (u i, s i ) 0. Tus [7, Lemma 3.] again gives te computable L (Ω) bound We ten conclude as in Case A via (6.). u i s i α i. 4

16 Teorem 6.3 (Guaranteed upper bound for te i-t eigenvalue). For a given i, let te approximate eigenvectors u k P p (T ), k i, be arbitrary, and let s k, k i, be teir eigenvector reconstructions by Definition 3.3. Suppose tat s k, k i, are linearly independent. Ten were ξ = i k= ξ k. In particular λ i Proof. Te statement follows immediately from te min max principle i k= max ξ ks k ξ R i, ξ = i k= ξ, (6.4) ks k λ s s = s. (6.5) λ i = min max v V i V, dimv i=i v V i v. 6. Eigenvectors We now turn to te estimates on te error in te approximation of te i-t exact eigenvector u i by u i and teir efficiency and robustness wit respect to te polynomial degree of u i. Teorem 6.4 (Guaranteed bounds for te i-t eigenvector error). Let te assumptions of Teorem 6. be satisfied. Ten te energy eigenvector error can be bounded via θ (u i u i ) η i + θ (u i s i ), (6.6) were η i is defined in te tree cases A, B, and C respectively by (6.5), (6.7), and (6.9). Proof. Te triangle inequality gives θ (u i u i ) θ (u i s i ) + θ (u i s i ). We conclude by te bounds θ (u i s i ) = (u i s i ) η i sown in (6.) and (6.3) in te proof of Teorem 6.. Teorem 6.5 (Efficiency and robustness of te estimates). Let te assumptions of Teorem 6. be satisfied. Ten estimate (6.6) is efficient as η i + θ (u i s i ) C i ( θ (u i u i ) + { e Π 0 e [u i ] e e E ) + u i + λi θ u i, } (6.7) were te generic constants C i can be determined from te detailed estimates: efficiency of θ (u i s i ) : θ (u i s i ) θ (u i s i ) + s i s i s i, θ (u i s i ) (u i s i ) + θ { d+c inv e Π 0 e [u i ] e e E (6.8a) }, (6.8b) togeter wit (4.7), inequalities (6.) and (6.9) below, and (u i u i ) θ (u i u i ) + θ { } d+c inv e Π 0 e [u i ] e ; (6.8c) e E 5

17 efficiency of u i s i (first part of η i,res ): u i s i C bf (u i s i ) + u i, Ω Ω togeter wit (4.7) and (6.8c); e E ext e efficiency of s i + σ i (second part of η i,res ): e E int e Π 0 e [u i ] e Π 0 e [u i ] e + u i, Ω, (6.9a), (6.9b) s i + σ i θ u i + σ i + θ (u i s i ), ( λ i θ u i + σ i (d + )C st C cont,pf u i s i λ (6.0a) (6.0b) + θ (u i s i ) + Res θ (s i, λ i ) ), Res θ (s i, λ i ) s i λi λ i λ i + C i s i (u i s i ), (6.0c) C i := if i =, { (λi ) C i := max, } if i >, λ λ i λ i θ (u i u i ) + θ (u i u i ) θ u i (6.0d) + λi θ u i, togeter wit (u i s i ) θ (u i u i ) + θ (u i s i ), (6.8), (6.9), and (6.) below; inequalities for te scaling terms: s i u i + u i s i, (6.a) s i θ u i + θ (u i s i ), u i u i s i s i u i + u i s i ; (6.b) (6.c) note tat α i given by (6.8a) only contains terms treated above (possibly wit multiplicative factors). Proof. We first examine te second term on te rigt-and side of (6.6). Te definition s i := te triangle inequality give (6.8a), since (s i s i ) = s i s i s i. s i s i and 6

18 As for te first term terein, θ (u i s i ) = (u i s i ) θ l e ([u i ]) e E (u i s i ) + θ l e ([u i ]) K T e E K K { (u i s i ) + θ (d + ) l e ([u i ]) K K T e E K = (u i s i ) + θ {(d + ) e E l e ([u i ]) Te }, } a direct consequence of te definition of te discrete gradient (.5), of te triangle and Caucy Scwarz inequalities, and of te fact tat l e ([u i ]) is only supported on te ( or ) elements in T e containing te face e. Next, te definition of te face lifting l e from Section.3, te fact tat v n e are constants for v V 0 (T e ), and te inverse inequality (5.4) give l e ([u i ]) Te = sup (l e ([u i ]), v ) Te v V 0 (T e); v Te = = sup {v } n e, Π 0 e [u i ] e v V 0 (T e); v Te = C inv e Π 0 e [u i ] e. Combining te two above bounds gives (6.8b). Finally, (6.8c) follows by, using again (.5), (u i u i ) = θ(u i u i ) θ l e ([u i ]) e E and proceeding as above for te liftings. Concerning te second term in (6.8a), te multiplicative factor s i approaces θ u i λ i as manifested in (6.b), te multiplicative factor s i is of order wen u i as sown in (6.c), and s i is bounded in (6.a) by te consistency term u i and te estimator u i s i efficient via (6.9). Tus te efficiency for te term θ (u i s i ) as announced in (6.7) follows. We now turn to te L (Ω)-term u i s i, te first part of te estimator η i,res given by (6.). Note tat η i,res forms te principal part of η i in all tree cases A, B, and C, and tat te scaling factor / s i is of order, see (6.c). First, (6.9a) is a consequence of te broken Poincaré Friedrics inequality (.4) wit v = u i s i and of te fact tat te jumps of s i are zero. Te last term in (6.9a) can ten still be bounded by u i, Ω = Π 0 e [u i ], e e Π 0 e [u i ] e e, e E ext so tat (6.9b) follows by anoter Caucy Scwarz inequality. Te efficiency of u i s i is ten completed by (4.7) and (6.8c). Numerically, toug, we ave observed tat u i s i converges still one order faster tan wat (6.9) suggests, so tat it becomes negligible in practice, see Remark 6.9 below. We now turn to te second part of te estimator η i,res of (6.), s i + σ i. To begin wit, (6.0a) follows by te triangle inequality; te second term terein as been treated in (6.8). For θ u i + σ i, we ave te crucial bound (4.5b). As, owever, u i H 0 (Ω), we need to get back from Res θ (u i, λ i ) to Res θ (s i, λ i ) to prove te efficiency. For tis purpose, let v V wit v = be fixed. Using te residual definition (.7a), te Caucy Scwarz and Poincaré Friedrics (.) inequalities, and te dual e E ext 7

19 norm definition (.7b), Res θ (u i, λ i ), v V,V = λ i (u i, v) ( θ u i, v) = λ i (u i s i, v) ( θ (u i s i ), v) + Res θ (s i, λ i ), v V,V λ i u i s i + θ (u i s i ) + Res θ (s i, λ i ), λ so tat (6.0b) follows. We know from (6.8) and (6.9) tat te first two terms erein are efficient, so tat we pursue wit te last one only. To start wit, note tat Res θ (s i, λ i ) = s i Res θ ( s i, λ i ) ; ten (6.0c) follows from te proof of te second bound in [7, Teorem 3.5]. To finis, develop λ i λ i = λ i θ (u i u i + u i ) = λ i θ (u i u i ) ( θ (u i u i ), θ u i ) θ u i, wic proves (6.0d) and togeter wit (6.8), (6.9), and (6.) gives te requested efficiency. 6.3 Comments We now give some comments on te results of Teorems ; a discussion for te conforming setting can be found in [7, Section 5.3]. Remark 6.6 (Vanising consistency terms). Nonconforming finite elements are a particular example of a numerical metod were bot consistency terms u i and λ i θ u i are zero and tus vanis in (6.7), see Section 7. below. Remark 6.7 (Jumps of mean value zero). A particular situation arises wen [u i ], e = 0 for all e E, i.e., wen te jumps over te mes faces in te eigenvector approximation vanis in mean value. Ten te discrete and broken gradient coincide, i.e., θ = (see Section.3) and all te mean value jump terms of te form e Π 0 e [u i ] e of te present paper vanis, in particular in (6.8a) and in (6.7). Moreover, (4.7) and (6.9a) simplify respectively to (u i s i ) (d + )C st C cont,bpf (u i u i ), u i s i (d + )C st C cont,bpf C bf (u i u i ), (6.a) (6.b) see [38, Lemma 3. and Section 4.3.]. Tis very favorable context arises namely for nonconforming and mixed finite elements, as we will see below in Sections 7. and 7.3. Remark 6.8 (Jump-free estimators for te symmetric discontinuous Galerkin metod). Te jump terms in te estimator α i given by (6.8a) also vanis wen θ =, wic is typically te situation for te symmetric discontinuous Galerkin metod of Section 7. below. Remark 6.9 (Alternative mean-value-preserving eigenvector reconstruction). Te term λ i λ u i s i in (4.5a), (6.), and (6.8a) wit te eigenvector reconstruction s i of Definition 3.3 typically converges by one order faster tan s i + σ i, wile allowing to prove polynomial degree robustness. One could additionally impose s i to be elementwise of te same mean value as u i, see [36, (3.) and (3.6)]. Ten λ λ i u i s i can be replaced by λ { i π K T ( K u i s i K ) }, wit an additional mes power. Indeed, in tis case λ i (s i u i, v) = λ i K T (s i u i, v Π 0 v) K λ i K T ( s i u i Kπ K v K ) in te proof of Corollary 4., by te Poincaré inequality and convexity of simplices. Remark 6.0 (Alternative eigenvector reconstruction and vanising jumps for te symmetric discontinuous Galerkin metod). An alternative eigenvector reconstruction ( ( to tat ) of Definition ) 3.3 is possible in two space 0 dimensions following [38, Remark 3.] wen θ u i, ψ 0 a = 0 for all a V. Tis is in ω a particular satisfied for te symmetric variant of te discontinuous Galerkin metod of Section 7. below. Tis alternative reconstruction remarkably yields θ (u i s i ) (d + )C st C cont,p θ (u i u i ) 8