Convergence of adaptive discontinuous Galerkin methods (corrected version of [Math. Comp. 87 (2018), no. 314, 2611--2640])
CCONVERGENCE OF ADAPTIVEDISCONTINUOUS GALERKIN METHODS (CORRECTED VERSION OF [MATH. COMP. 87 (2018), NO. 314, 2611–2640])
CHRISTIAN KREUZER AND EMMANUIL H. GEORGOULIS
Abstract.
We develop a general convergence theory for adaptive discontinu-ous Galerkin methods for elliptic PDEs covering the popular SIPG, NIPG andLDG schemes as well as all practically relevant marking strategies. Anotherkey feature of the presented result is, that it holds for penalty parameters onlynecessary for the standard analysis of the respective scheme. The analysisis based on a quasi interpolation into a newly developed limit space of theadaptively created non-conforming discrete spaces, which enables to generalisethe basic convergence result for conforming adaptive finite element methods byMorin, Siebert, and Veeser [A basic convergence result for conforming adaptivefinite elements, Math. Models Methods Appl. Sci., 2008, 18(5), 707–737]. Introduction
Discontinuous Galerkin finite element methods (
DGFEM ) have enjoyed consid-erable attention during the last two decades, especially in the context of adaptivealgorithms (
ADGM s): the absence of any conformity requirements across elementinterfaces characterizing
DGFEM approximations allows for extremely general adap-tive meshes and/or an easy implementation of variable local polynomial degrees inthe finite element spaces. There has been a substantial activity in recent years forthe derivation of a posteriori bounds for discontinuous Galerkin methods for ellipticproblems [KP03, BHL03, Ain07, HSW07, CGJ09, EV09, ESV10, ZGHS11, DPE12].Such a posteriori estimates are an essential building block in the context of adaptivealgorithms, which typically consist of a loop(1.1)
SOLVE Ñ ESTIMATE Ñ MARK Ñ REFINE . The convergence theory, however, for the ‘extreme’ non-conformity case of
ADGM shad been a particularly challenging problem due to the presence of a negative powerof the mesh-size h stemming from the discontinuity-penalization term. As a conse-quence, the error is not necessarily monotone under refinement. Indeed, consultingthe unprecedented developments of convergence and optimality theory of conform-ing adaptive finite element methods ( AFEM s) during the last two decades, the strictreduction of some error quantity appears to be fundamental for most of the results.In fact, D¨orfler’s marking strategy typically ensures that the error is uniformly re-duced in each iteration [D¨or96, MNS00, MNS02] and leads to optimal convergencerates [Ste07, CKNS08, KS11, DK08, BDK12]; compare also with the monographs[NSV09, CFP14] and the references therein. Showing that the error reduction isproportional to the estimator on the refined elements, instance optimality of anadaptive finite element method was shown recently for an
AFEM with modifiedmarking strategy in [DKS16, KS16]. A different approach was, however, taken in
Date : September 7, 2020.
Mathematics Subject Classification.
Primary 65N30, 65N12, 65N50, 65N15.
Key words and phrases.
Adaptive discontinuous Galerkin methods, convergence, ellipticproblems.The research of Christian Kreuzer was supported by the DFG research grant KR 3984/5-1.Emmanuil H. Georgoulis acknowledges support by the Leverhulme Trust. a r X i v : . [ m a t h . NA ] S e p CH. KREUZER AND E. H. GEORGOULIS [MSV08, Sie11], where convergence of the
AFEM is proved, exploiting that the ap-proximations converge to a solution in the closure of the adaptively created finiteelement spaces in the trial space together with standard properties of the a posteri-ori bounds. The result covers a large class of inf-sup stable PDEs and all practicallyrelevant marking strategies without yielding convergence rates though.Karakashian and Pascal [KP07] gave the first proof of convergence for an adaptive
DGFEM based on a symmetric interior penalty scheme (
SIPG ) with D¨orfler markingfor Poisson’s problem. Their proof addresses the challenge of negative power of h inthe estimator, by showing that the discontinuity-penalization term can be controlledby the element and jump residuals only, provided that the DGFEM discontinuity-penalisation parameter, henceforth denoted by σ , is chosen to be sufficiently large ;the element and jump residuals involve only positive powers of h and, therefore, canbe controlled similarly as for conforming methods. The optimality of the adaptive SIPG was shown in [BN10]; see also [HKW09].The standard error analysis of the
SIPG requires that σ is sufficiently large forthe respective bilinear from to be coercive with respect to an energy-like norm. Itis not known in general, however, whether the choice of σ required for coercivityof the interior penalty DGFEM bilinear form is large enough to ensure that thediscontinuity-penalization term can be controlled by the element and jump residualsonly. Therefore, the convergence of
SIPG is still open for values of σ large enoughfor coercivity but, perhaps, not large enough for the crucial result from [KP07] tohold. To the best of our knowledge, the only result in this direction is the proof ofconvergence of a weakly overpenalized ADGM for linear elements [GG14], utilizingthe intimate relation between this method and the lowest order Crouzeix-Raviartelements.This work is concerned with proving that the
ADGM converges for all values of σ for which the method is coercive, thereby settling the above discrepancy between themagnitude of σ required for coercivity and the, typically much larger, values requiredfor proof of convergence of ADGM . Apart from settling this open problem theoret-ically, this new result has some important consequences in practical computations:it is well known that as σ grows, the condition number of the respective stiffnessmatrix also grows. Therefore, the magnitude of the discontinuity-penalization pa-rameter σ affects the performance of iterative linear solvers, whose complexity isalso typically included in algorithmic optimality discussions of adaptive finite ele-ments. In addition, the theory presented here includes a large class of practicallyrelevant marking strategies and covers popular discontinuous Galerkin methods likethe local discontinuous Galerkin method (LDG) and even the nonsymmetric inte-rior penalty method ( NIPG ), which are coercive for any σ ą
0. Moreover, we expectthat it can be generalised to non-conforming discretisations for a number of otherproblems like the Stokes equations or fourth order elliptic problems. However, asfor the conforming counterpart [MSV08], no convergence rates are guaranteed.The proof of convergence of the
ADGM , discussed below, is motivated by thebasic convergence for the conforming adaptive finite element framework of Morin,Siebert and Veeser [MSV08]. More specifically, we extend considerably the ideasfrom [MSV08] and [Gud10] to be able to address the crucial challenge that the limitsof
DGFEM solutions, constructed by the adaptive algorithm, do not necessarilybelong to the energy space of the boundary value problem as well as to concludeconvergence from a perturbed best approximation result.To highlight the key theoretical developments without the need to resort to com-plicated notation, we prefer to focus on the simple setting of the Poisson problemwith essential homogeneous boundary conditions and conforming shape regular tri-angulations. We believe, however, that the results presented below are valid forgeneral elliptic PDEs including convection and reaction phenomena as well as forsome classes of non conforming meshes; compare with [BN10].
ONVERGENCE OF
ADGM The remainder of this work is structured as follows. In Section 2 we shall in-troduce the
ADGM framework for Poisson’s equation and state the main result,which is then proved in Section 5 after some auxiliary results, needed to generalise[MSV08], are provided in Sections 3 and 4. In particular, in Section 3 a space ispresented, which is generated from limits of discrete discontinuous functions in thesequence of discontinuous Galerkin spaces constructed by
ADGM . Section 4 is thenconcerned with proving that the sequence of discontinuous Galerkin solutions pro-duced by
ADGM converges indeed to a generalised Galerkin solution in this limitspace. This follows from an (almost) best-approximation property, generalising theideas in [Gud10]. 2.
The ADGM and the main result
Let a measurable set ω and a m P N . We consider the Lebesgue space L p ω ; R m q of square integrable functions over ω with values in R m , with inner product x¨ , ¨y ω and associated norm }¨} ω . We also set L p ω q : “ L p ω ; R q . The Sobolev space H p ω q is the space of all functions in L p ω q whose weak gradient is in L p ω ; R d q , for d P N .Thanks to the Poincar´e-Friedrichs’ inequality, the closure H p ω q of C p ω q in H p ω q is a Hilbert space with inner product x ∇ ¨ , ∇ ¨y ω and norm } ∇ ¨} ω . Also, we denotethe dual space H ´ p ω q of H p ω q , with the norm } v } H ´ p ω q : “ sup w P H p ω q x v , w y} ∇ w } ω , v P H ´ p ω q , with dual brackets defined by x v , w y : “ v p w q , for w P H p ω q .Let Ω Ă R d , d “ ,
3, be a bounded polygonal ( d “
2) or polyhedral ( d “ ´ ∆ u “ f in Ω , u “ B Ω , with f P L p Ω q . The weak formulation of (2.1) reads: find u P H p Ω q , such that x ∇ u, ∇ v y Ω “ x f, v y Ω for all v P H p Ω q . (2.2)From the Riesz representation theorem, it follows that the solution u exists and isunique.2.1. Discontinuous Galerkin method.
Let G be a conforming (that is, not con-taining any hanging nodes) subdivision of Ω into disjoint closed simplicial elements E so that ¯Ω “ Ť t E : E P G u and set h E : “ | E | { d . Let S “ S p G q be the setof p d ´ q -dimensional element faces S associated with the subdivision G including B Ω, and let ˚ S “ ˚ S p G q Ă S by the subset of interior faces only. We also introducethe mesh size function h G : Ω Ñ R , defined by h G p x q : “ h E , if x P E zB E and h G p x q “ h S : “ | S | {p d ´ q , if x P S P S and set Γ “ Γ p G q “ Ť t S : S P S u and ˚Γ “ ˚Γ p G q “ Ť t S : S P ˚ S u . We assume that G is derived by iterative or recursive newestvertex bisection of an initial conforming mesh G ; see [B¨an91, Kos94, Mau95, Tra97].We denote by G the family of shape regular triangulations consisting of such sub-divisions of G .Let P r p E q denote the the space of all polynomials on E of degree at most r P N ,we define the discontinuous finite element space(2.3) V p G q : “ ź E P G P r p E q Ă ź E P G W ,p p E q “ : W ,p p G q , ď p ď 8 , and H p G q : “ W , p G q . Let N “ N p G q be the set of Lagrange nodes of V p G q anddefine the neighbourhood of a node z P N p G q by N G p z q : “ t E P G : z P E u ,and the union of its elements by ω G p z q “ Ť t E P G : z P E u . We also definethe corresponding neighbourhoods for all elements E P G by N G p E q : “ t E P G : E X E ‰ Hu and ω G p E q “ Ť t E P G : E X E ‰ Hu “ Ť t ω G p z q : z P N p E q X E u ,respectively, and set ω G p S q : “ Ť t E P G : S Ă E u ; compare with Figure 1. Thenumbers of neighbours N G p z q and N G p E q are uniformly bounded for all z P N ,respectively E P G , depending on the shape regularity of G and, thus, on G . CH. KREUZER AND E. H. GEORGOULIS E Figure 1.
The neighbourhood N G p E q of some E P G .Let E ` , E ´ be two generic elements sharing a face S : “ E ` X E ´ P ˚ S and let n ` and n ´ the outward normal vectors of E ` respectively E ´ on S . For q : Ω Ñ R and φ : Ω Ñ R d , let q ˘ : “ q | S XB E ˘ and φ ˘ : “ φ | S XB E ˘ , and set tt q uu| S : “ p q ` ` q ´ q , tt φ uu| S : “ p φ ` ` φ ´ q , rr q ss | S : “ q ` n ` ` q ´ n ´ , rr φ ss | S : “ φ ` ¨ n ` ` φ ´ ¨ n ´ ;if S Ă B E X B
Ω, we set tt φ uu| S : “ φ ` and rr q ss | S : “ q ` n ` .In order to define the discontinuous Galerkin schemes, we introduce the followinglocal lifting operators. For S P S , we define R S G : L p S q d Ñ ś E P G P (cid:96) p E q d and L S G : L p S q Ñ ś E P G P (cid:96) p E q d by ˆ Ω R S G p φ q ¨ τ d x “ ˆ S φ ¨ tt τ uu d s @ τ P ź E P G P (cid:96) p E q d (2.4a)and ˆ Ω L S G p q q ¨ τ d x “ ˆ S q rr τ ss d s @ τ P ź E P G P (cid:96) p E q d , (2.4b)with (cid:96) P t r, r ` u . Note that L S G p q q and R S G p φ q vanish outside ω G p S q . Moreover,using the local definition and the boundedness of the lifting operators in a referencesituation together with standard scaling arguments, we have for φ P P r p S q d and q P P r p S q that ›› L S G p φ q ›› Ω À ››› h ´ { G φ ››› S and ›› R S G p q q ›› Ω À ››› h ´ { G q ››› S ;(2.4c)compare with [ABCM02]. Also, here and below we write a À b when a ď Cb for aconstant C not depending on the local mesh size of G or other essential quantitiesfor the arguments presented below. Observing that the sets ω G p S q , S P S do overlapat most d ` R G : L p Γ q d Ñ V p G q d and L G : L p ˚Γ q Ñ V p G q d defined by R G p φ q : “ ÿ S P S R S G p φ q and L G p q q : “ ÿ S P ˚ S R S G p q q , that } R G prr v ssq} Ω À ››› h ´ { G v ››› Γ and } L G p β ¨ rr v ssq} Ω À | β | ››› h ´ { G v ››› ˚Γ for all v P V p G q and β P R d . ONVERGENCE OF
ADGM We define the bilinear form B G r¨ , ¨s : V p G q ˆ V p G q Ñ R by B G r w, v s : “ ˆ G ∇ w ¨ ∇ v d x ´ ˆ S ` tt ∇ w uu ¨ rr v ss ` θ tt ∇ v uu ¨ rr w ss ˘ d s ` ˆ ˚ S ` β ¨ rr w ss rr ∇ v ss ` rr ∇ w ss β ¨ rr v ss ˘ d s ` ˆ Ω γ ` R G prr w ssq ` L G p β ¨ rr w ssq ˘ ¨ ` R G prr v ssq ` L G p β ¨ rr v ssq ˘ d x ` ˆ S σh G rr w ss ¨ rr v ss d s ;(2.5)for θ P t˘ u , γ P t , u , β P R d and σ ě
0. Here we have used the short-handnotation ˆ G ¨ d x : “ ÿ E P G ˆ E ¨ d x and ˆ S ¨ d s : “ ÿ S P S ˆ S ¨ d s. We consider the choices θ “ β “ , and γ “ SIPG ) [DD76], θ “ ´ β “ , and γ “ NIPG ) [RWG99], and θ “ β P R d , and γ “ LDG ) [CS98]; compare alsowith [ABCM02] and [JNS16].In all three cases, the corresponding discontinuous Galerkin finite element method ( DGFEM ) then reads: find u G P V p G q such that(2.6) B G r u G , v G s “ ˆ Ω f v G d x “ : l p v G q , for all v G P V p G q . Upon denoting by ∇ pw v the piecewise gradient ∇ pw v | E “ ∇ v | E for all E P G , thecorresponding energy norm |||¨||| G is defined by ||| w ||| G : “ ´ ›› ∇ pw w ›› ` ¯ σ ››› h ´ { G rr w ss ››› ¯ { , for w | E P H p E q , E P G . Here ¯ σ : “ max t , σ u . Also, for some subset M Ă G with ω “ Ť t E | E P M u , we define ||| w ||| M : “ ´ ›› ∇ pw w ›› ω ` ¯ σ ››› h ´ { G rr w ss ››› p M q ¯ { . If for
SIPG we have σ : “ C σ r for some constant C σ ą σ ą NIPG and for
LDG σ ą (cid:96) “ r and σ “ (cid:96) “ r ` α “ α p σ q ą
0, such that α ||| w ||| G ď B G r w, w s @ w P H p G q , (2.7)i.e. all three DGFEM s are coercive in V p G q . Note that coercivity (2.7) holds truealso for functions in H p G q after extending the discrete bilinear form using someliftings; see, e.g., [Arn82, ABCM02, JNS16] for details. The choice ¯ σ “ max t , σ u accounts for the fact that we can have σ “ Proposition 1 (Poincar´e- V p G q ) . Let G be a triangulation of Ω and G ‹ some re-finement of G . Then, for v P V p G ‹ q , E P G and v E : “ | ω G p E q| ´ ´ ω G p E q v d x , wehave } v ´ v E } ω G p E q À ˆ ω G p E q h G | ∇ pw v | d x ` ˆ S P S ‹ ,S Ă ω G p E q h G h ´ G ‹ rr v ss d s, where S ‹ “ S p G ‹ q and the hidden constant depends on d and on the shape regularityof N G p E q . CH. KREUZER AND E. H. GEORGOULIS
The next important result from [KP03, Theorem 2.2] (compare also with [BN10,Lemma 6.9] and [BO09, Theorem 3.1]) quantifies the local distance of a discretenon-conforming function to the conforming subspace with the help of the of thescaled jump terms.
Proposition 2.
For G P G , there exists an interpolation operator I G : H p G q Ñ V p G q X H p Ω q , such that we have ››› h ´ { G p v ´ I G v q ››› E ` } ∇ p v ´ I G v q} E À ˆ B E h ´ G rr v ss d s, for all E P G and v P V p G q . From this, we can easily deduce the following broken Friedrichs type inequality;compare also with [BO09, (4.5)].
Corollary 3 (Friedrichs- V p G q ) . Let G P G , then } v } Ω À ||| v ||| G for all v P V p G q . Let BV p Ω q denote the Banach space of functions with bounded variation equipedwith the norm } v } BV p Ω q “ } v } L p Ω q ` | Dv |p Ω q , where Dv is the measure representing the distributional derivative of v with totalvariation | Dv |p Ω q “ sup φ P C p Ω q d , } φ } L Ω qď ˆ Ω v div φ d x. Here the supremum is taken over the space C p Ω q d of all vector valued continuouslydifferentiable functions with compact support in Ω.Another crucial result [BO09, Lemma 2] states then that the total variation of thedistributional derivative of broken Sobolev functions is bounded by the discontiuousGalerkin norm. Proposition 4.
For G P G we have that | Dv |p Ω q À } v } L p Ω q ` ˆ S |rr v ss| d s À ||| v ||| G for all v P H p G q . A posteriori error bound.
We recall the a posteriori results from [KP03,BN10, BGC05, BHL03]; compare also with [CGJ09].For v P V p G q , we define the local error indicators for E P G by E G p v, E q : “ ´ ˆ E h G | f ` ∆ v | d x ` ˆ B E X Ω h G rr ∇ v ss d s ` σ ˆ B E h ´ G rr v ss d s ¯ { ;when v “ u G , we shall write E G p E q : “ E G p u G , E q . Also, for M Ă G , we set E G p v, M q : “ ´ ÿ E P M E p v, E q ¯ { . Proposition 5.
Let u P H p Ω q be the solution of (2.2) and u G P V p G q its respective DGFEM approximation (2.6) on the grid G P G . Then, ||| u ´ u G ||| G À ÿ E P G E G p E q . The efficiency of the estimator follows with the standard bubble function tech-nique of Verf¨urth [Ver96, Ver13]; compare also with [KP03, Theorem 3.2], [Gud10,Lemma 4.1] and Proposition 23 below.
ONVERGENCE OF
ADGM Proposition 6.
Let u P H p Ω q be the solution of (2.2) and let G P G . Then, forall v P V p G q and E P G , we have ˆ E h G | f ` ∆ v | d x ` ˆ B E X Ω h G rr ∇ v ss d s À } u ´ v } ω G p E q ` ›› ∇ pw p u ´ v q ›› ω G p E q ` osc p N G p E q , f q , with data-oscillation defined by osc p M , f q : “ ´ ÿ E P M osc p E, f q ¯ { , where osc p E, f q : “ inf f E P P r ´ } h G p f ´ f E q} E , for all M Ă G . In particular, this implies E G p v, E q À ||| v ´ u ||| N G p E q ` osc p N G p E q , f q . Remark 7.
Note that the presented theory obviously applies to all locally equivalentestimators as well; compare e.g. with [KP03, BN10, BGC05, BHL03, CGJ09] . Forthe sake of a unified presentation, we restrict ourselves to the above representation.
Adaptive discontinuous Galerin finite element method (
ADGM ). Theadaptive algorithm, whose convergence will be shown below, reads as follows.
Algorithm 8 ( ADGM ) . Starting from an initial triangulation G , the adaptivealgorithm is an iteration of the following form(1) u k “ SOLVE p V p G k qq ;(2) t E k p E qu E P G k “ ESTIMATE p u k , G k q ;(3) M k “ MARK ` t E k p E qu E P G k , G k ˘ ;(4) G k ` “ REFINE p G k , M k q ; increment k .Here we have used the notation E k p E q : “ E G k p E q , for brevity. SOLVE . We assume that the output u G “ SOLVE p V p G qq is the DGFEM approximation (2.6) of u with respect to V p G q . ESTIMATE . We suppose that t E G p E qu E P G : “ ESTIMATE p u G , G q computes the error indicators from Section 2.2. MARK . We assume that the output M : “ MARK pt E G p E qu E P G , G q of marked elements satisfies E G p E q ď g p E G p M qq , for all E P G z M . (2.8)Here g : R ` Ñ R ` is a fixed function, which is continuous in 0 with g p q “
0, i.e.lim (cid:15) Ñ g p (cid:15) q “ REFINE . We assume for M Ă G P G , that for the refined grid˜ G : “ REFINE p G , M q we have E P M ñ E P G z ˜ G , (2.9)i.e., each marked element is refined at least once. CH. KREUZER AND E. H. GEORGOULIS
Figure 2.
Selection of a sequence of triangulations of Ω “ p , q ,where in each iteration the elements in Ω ´ “ r , . s ˆ r , . s aremarked for refinement. The elements G ` in the remaining domainΩ z Ω ´ are, after some iteration, not refined anymore. Moreover,after some iteration, their whole neighbourhood is not refined any-more.2.4. The main result.
The main result of this work states that the sequence ofdiscontinuous Galerkin approxiations, produced by
ADGM , converges to the exactsolution of (2.1).
Theorem 9.
We have that E k p G k q Ñ as k Ñ 8 . In particular, this implies that ||| u ´ u k ||| k Ñ as k Ñ 8 . A limit space and quasi-interpolation
In this section we shall first introduce a new limit space V of the sequenceof adaptively constructed discontinuous finite element spaces t V p G k qu k P N . A newquasi-interpolation operator is then introduced in Section 3.3 in order to to provethat there exists a unique Galerkin solution u of a generalised discontinuousGalerkin problem in V .3.1. Sequence of partitions.
The
ADGM produces a sequence t G k u k P N of nestedadmissible partitions of Ω. Following [MSV08], we define G ` : “ ď k ě č j ě k G j , and Ω ` : “ Ω p G ` q to be the set and domain of all elements, respectively, which eventually will not berefined any more; here Ω p X q : “ interior p Ť t E : E P X uq for a collection of elements X . We also define the complementary domain Ω ´ : “ Ω z Ω ` . For the ease ofpresentation, in what follows, we shall replace subscripts G k by k to indicate theunderlying triangulation, e.g. we write N k p E q instead of N G k p E q .The following result states that neighbours of elements in G ` are eventually alsoelements of G ` ; cf., [MSV08, Lemma 4.1]. Lemma 10.
For E P G ` there exists a constant K “ K p E q P N , such that N k p E q “ N K p E q for all k ě K, i.e., we have N k p E q Ă G ` for all k ě K . Next, for a fixed k P N , we set G ` k : “ G k X G ` , Ω ` k : “ Ω p G ` k q , G `` k : “ t E P G k : N k p E q Ă G ` u , Ω `` k : “ Ω p G `` k q , G ´ k : “ G k z G `` k Ω ´ k : “ Ω p G ´ k q ; ONVERGENCE OF
ADGM compare also with Figure 2. This notation is also adopted for the correspondingfaces, e.g., we denote S ` k : “ S p G ` k q and ˚ S ` k : “ ˚ S p G ` k q and correspondingly for allother above sub-triangulations of G k .The next lemma states that the meshsize of elements in G ´ k converges uniformlyto zero; compare also with [MSV08, (4.15) and Corollary 4.1] and [Sie11, Corollary3.3]. Lemma 11.
We have that lim k Ñ8 } h k χ Ω ´ k } L p Ω q “ , with χ Ω ´ k denoting the char-acteristic function of Ω ´ k .In particular, this implies that lim k Ñ8 | Ω p N k p G ´ k qqz Ω ´ | “ lim k Ñ8 | Ω ´ k z Ω ´ | “ .Proof. The first claim is proved in [Sie11, Corollary 3.3].In order to prove the second claim, we first observe that | Ω ´ k z Ω ´ | “ | Ω p G ` z G `` k q| . For (cid:96) P N , it follows from Lemma 10 and G ` (cid:96) ă 8 , that there exists K p (cid:96) q , suchthat G ` (cid:96) Ă G `` K p (cid:96) q and thus | Ω p G ` z G `` K p (cid:96) q q| ď | Ω p G ` z G ` (cid:96) q| Ñ (cid:96) Ñ 8 , (3.1)i.e. a subsequence of t| Ω p G ` z G `` k q|u k vanishes. However, since the sequence ismonotone decreasing, it must vanish as a whole. To conclude, we first realise that | Ω p N k p G ´ k qqz Ω ´ | ď | Ω ´ k z Ω ´ | ` | Ω p N k p G ´ k qz G ´ k q| ; it remains to prove that the latterterm vanishes as k Ñ 8 . To this end, we observe that N k p G ´ k qz G ´ k Ă G `` k z G ` k , with G ` k “ t E P G k : N k p E q P G `` k u . Indeed, assume that E P G ` k X N k p G ´ k q , thenthere exists E P N k p E q with E P G `` k X G ´ k ; this is a contradiction. Consequently,we have | Ω p N k p G ´ k qz G ´ k q| ď | Ω p G `` k z G ` k q| ď | Ω p G ` z G ` k q| Ñ k Ñ 8 , where the last limit follows by iterating the reasoning from (3.1). (cid:3) The limit space.
In this section, we shall investigate the limit of the finiteelement spaces V k : “ V p G k q , k P N . To this end, we define V : “ (cid:32) v P BV p Ω q : v | Ω ´ P H B Ω XB Ω ´ p Ω ´ q and v | E P P r @ E P G ` such that Dt v k u k P N , v k P V k with lim sup k Ñ8 ||| v k ||| k ă 8 and lim k Ñ8 ||| v ´ v k ||| k ` } v ´ v k } Ω “ ( ;here H B Ω XB Ω ´ p Ω ´ q denotes the space of functions from H p Ω q restricted to Ω ´ .Moreover, we have extended the definition of the piece-wise gradient to ∇ pw v P L p Ω q d : ∇ pw v | Ω ´ “ ∇ v | Ω ´ and ∇ pw v | E “ ∇ v | E @ E P G ` . (3.2)Note that for v P BV p Ω q there exists the L -trace of v on Γ k “ Ť t S : S P S k u ;compare e.g. with the trace theorem [BO09, Theorem 4.2]. In other words, v ismeasurable with respect to the p d ´ q -dimensional Hausdorff measure on S k and,therefore, the term ||| v ||| k , v P V , makes sense. Obviously, we have V k X C p Ω q Ă V for all k P N and, thus, V is not empty.Setting h ` : “ h G ` and S ` : “ S p G ` q , we define x v, w y : “ ˆ Ω ∇ pw v ¨ ∇ pw w d x ` ¯ σ ˆ S ` h ´ ` rr v ss rr w ss d s, and ||| v ||| : “ x v, v y { , for all v, w P V .We shall next list some basic properties of the space V . Proposition 12.
For v P V , we have ||| v ||| k Õ ||| v ||| ă 8 as k Ñ 8 . In particular, for fixed (cid:96) P N , let E P G (cid:96) ; then, we have ˆ t S P S k : S Ă E u h ´ k rr v ss d s Õ ˆ t S P S ` : S Ă E u h ´ ` rr v ss d s, as k Ñ 8 . Proof.
Since v P V , there exists t v k u k P N , v k P V k with lim k Ñ8 ||| v ´ v k ||| k “ k Ñ8 ||| v k ||| k ă 8 . We first observe that ||| v ||| k ď ||| v ´ v k ||| k ` ||| v k ||| k ă 8 uniformly in k . Thanks to the mesh-size reduction, i.e. h m ď h k for all m ě k , weconclude that ˆ S k h ´ k rr v ss d s ď ˆ S k h ´ m rr v ss d s ď ˆ S m h ´ m rr v ss d s, thanks to the inclusion Ť S P S k S Ă Ť S P S m S .Therefore, we have ||| v ||| k ď ||| v ||| m for all m ě k and, thus, t||| v ||| k u k P N converges.Consequently, for (cid:15) ą K “ K p (cid:15) q , such that for all k ě K and m ą k large enough, we have (cid:15) ą | ||| v ||| m ´ ||| v ||| k | “ ¯ σ ˆ S m zp S m X S k q h ´ m rr v ss d s ´ ¯ σ ˆ S k zp S m X S k q h ´ k rr v ss d s ě p {p d ´ q ´ q ¯ σ ˆ S k zp S m X S k q h ´ k rr v ss d s ě p {p d ´ q ´ q ¯ σ ˆ S k z S ` k h ´ k rr v ss d s. This follows from the fact that h m | S ď ´ {p d ´ q h k | S for all S P S k zp S m X S k q together with S ` k “ S m X S k for sufficiently large m ą k .Therefore, we have ´ S k z S ` k h ´ k rr v ss d s Ñ k Ñ 8 and, thus, ||| v ||| k “ ˆ Ω | ∇ pw v | d x ` ¯ σ ˆ S ` k h ´ k rr v ss d s ` ¯ σ ˆ S k z S ` k h ´ k rr v ss d s Ñ ||| v ||| ` . This proves the first claim. The second claim is a localised version and followscompletely analogously. (cid:3)
As a consequence, we have that Friedrichs and Poincar´e inequalities are inheritedto V . Corollary 13 (Friedrichs- V ) . We have } v } Ω À ||| v ||| for all v P V . Proof.
Since v P V , there exists t v k u k P N , v k P V k with lim sup k Ñ8 ||| v k ||| k ă 8 andlim k Ñ8 ||| v ´ v k ||| k ` } v ´ v k } Ω “
0. It thus follows from Corollary 3 and Proposi-tion 12 that } v } Ω “ lim k Ñ8 } v k } Ω À lim k Ñ8 ||| v k ||| k “ ||| v ||| . This is the assertion. (cid:3)
Lemma 14 (Poincar´e- V ) . Fix k P N and let E P G k . Then for v P V and v E : “ | ω k p E q| ´ ω k p E q v d x , we have } v ´ v E } ω k p E q À ›› h k ∇ pw v ›› ω k p E q ` ˆ t S P S ` : S Ă ω k p E qu h k h ´ ` rr v ss d s. ONVERGENCE OF
ADGM Proof.
By the definition of V , there exists v (cid:96) P V (cid:96) , (cid:96) P N , with lim (cid:96) Ñ8 ||| v ´ v (cid:96) ||| (cid:96) `} v ´ v (cid:96) } Ω “ (cid:96) Ñ8 ||| v (cid:96) ||| (cid:96) ă 8 . Therefore, we have ›› ∇ pw v (cid:96) ›› ω k p E q ` ˆ t S P S (cid:96) : S Ă ω k p E qu h ´ (cid:96) rr v (cid:96) ss d s Ñ ›› ∇ pw v ›› ω k p E q ` ˆ t S P S ` : S Ă ω k p E qu h ´ ` rr v ss d s as (cid:96) Ñ 8 ;see Proposition 12. Moreover, we have } v E ´ v (cid:96),E } ω k p E q ď } v ´ v (cid:96) } ω k p E q ď } v ´ v (cid:96) } Ω Ñ (cid:96) Ñ 8 , where v (cid:96),E : “ | ω k p E q| ´ ω k p E q v (cid:96) d x . We conclude with Proposition 1 that } v ´ v E } ω k p E q Ð } v (cid:96) ´ v (cid:96),E } ω k p E q À ›› h k ∇ pw v (cid:96) ›› ω k p E q ` ˆ t S P S (cid:96) : S Ă ω k p E qu h k h ´ (cid:96) rr v (cid:96) ss d s Ñ ›› h k ∇ pw v ›› ω k p E q ` ˆ t S P S ` : S Ă ω k p E qu h k h ´ ` rr v ss d s, as (cid:96) Ñ 8 . (cid:3) In order to extend the dG bilinear form (2.5) to V , we need to define appropriatelifting operators. For each S P S ` , there exists (cid:96) “ (cid:96) p S q P N , such that S P S `` (cid:96) . Wedefine the local lifting operators R S : L p S q d Ñ L p Ω q d and L S : L p S q Ñ L p Ω q d by R S “ R S(cid:96) : “ R S G (cid:96) and L S “ L S(cid:96) : “ L S G (cid:96) . (3.3)From (2.4) it is easy to see, that R S(cid:96) and L S(cid:96) depend only on S and the at mosttwo adjacent elements E, E P G ` (cid:96) with S Ă E X E . Therefore, and thanks to thefact that the G ` k are nested, we have that R S(cid:96) “ R Sk for all k ě (cid:96) and, thus, thedefinition is unique. We formally define the global lifting operators by R : “ ÿ S P S ` R S and L : “ ÿ S P ˚ S ` L S ;here ˚ S ` : “ t S P S ` : S R B Ω u .Moreover, from the local estimates (2.4c), it is easy to see that for v P V and β P R d , we have that ř S P S ` k R S prr v ssq and ř S P ˚ S ` k L S p β ¨rr v ssq are Cauchy sequencesin L p Ω q d . Consequently, R prr v ssq , L p β ¨ rr v ssq P L p Ω q are well posed and we have } R prr v ssq} Ω À ››› h ´ { ` v ››› Γ ` and } L p β ¨ rr v ssq} Ω À | β | ››› h ´ { ` v ››› ˚Γ ` , (3.4)where Γ ` “ Ť t S : S P S ` u and ˚Γ ` “ Ť t S : S P ˚ S ` u . This enables us to generalisethe discontinuous Galerkin bilinear form to V setting B r w, v s : “ ˆ Ω ∇ pw w ¨ ∇ pw v d x ´ ˆ S ` ` tt ∇ w uu ¨ rr v ss ` θ tt ∇ v uu ¨ rr w ss ˘ d s ` ˆ ˚ S ` ` β ¨ rr w ss rr ∇ v ss ` rr ∇ w ss β ¨ rr v ss ˘ d s ` ˆ Ω γ ` R prr w ssq ` L p β ¨ rr w ssq ˘ ¨ ` R prr v ssq ` L p β ¨ rr v ssq ˘ d x ` ˆ S ` σh ` rr w ss ¨ rr v ss d s, for v, w P V . Lemma 15.
The space ` V , x¨ , ¨y ˘ is a Hilbert space. Corollary 16.
There exists a unique u P V , such that B r u , v s “ ˆ Ω f v d x for all v P V . (3.5)In order to prove the last two statements, we introduce a new quasi-interpolation,which is designed in due consideration of the future refinements. The proofs ofLemma 15 and Corollary 16 are postponed to the end of Section 3.3.3.3. Quasi-interpolation.
We shall now define a quasi-interpolation operator Π k ,which maps into V X V k ; this will be a key technical tool in the analysis. On theone hand, membership in V X V k suggests to use some Cl´ement type interpolationsince the mapped functions need to be continuous in Ω ´ . On the other hand,the fact that the ADGM may leave some elements (namely G ` k Ą G `` k ) unrefined,suggests to define Π k to be the identity on these elements. Note that the quasi-interpolation operator from [CGS13] is motivated by a similar idea in order to mapfrom one Crouzeix-Raviart space into its intersection with a finer one.For fixed k P N , let t Φ Ez : E P G k , z P N k p E qu be the Lagrange basis of V k : “ V p G k q , i.e., Φ Ez is a piecewise polynomial of degree r with supp p Φ Ez q “ E andΦ Ez p y q “ δ zy for all z, y P N k . Its dual basis is then the set t Ψ Ez : E P G k , z P N k p E qu of piecewise polynomials ofdegree r , such that supp p Ψ Ez q “ E and @ Ψ Ey , Φ Ez D Ω “ δ zy for all z, y P N k p E q . For all (cid:96) ě k , we define Π k : L p Ω q Ñ L p Ω q byΠ k v : “ ÿ E P G k ÿ z P N k p E q p Π k v q| E p z q Φ Ez , (3.6)where for z P N k p E q we have that p Π k v q| E p z q : “ $’&’% ´ E v Ψ Ez d x, if N k p z q X G `` k ‰ H , else if z P B Ω ř E P N k p z q | E || ω k p z q| ´ E v Ψ E z d x, else.(3.7)Beyond standard stability and interpolation estimates for H p Ω q functions [SZ90,DG12], we list the following properties related to our setting. Lemma 17 (Properties of Π k ) . The operator Π k : L p Ω q Ñ L p Ω q defined in (3.6) has the following properties:(1) Π k : L p p Ω q Ñ L p p Ω q is a linear and bounded projection for all ď p ď 8 .In particular, we have that } Π k v } L p p E q À } v } L p p ω k p E qq , where the constant solely depends on p , r , d , and the shape regularity of G .(2) Π k v P V k for all v P L p Ω q ;(3) Π k v | E “ v | E , if E P G k and v | ω k p E q P P r p ω k p E qq ;(4) Π k v | E “ v | E , if E P G `` k and v | E P P r p E q ; if moreover v P V k , then also rr v ´ Π k v ss | S ” for all S P S `` k .(5) Π k v | Ω z Ω ` k P C p Ω z Ω ` k q and rr Π k v ss “ on Bp Ω z Ω ` k q ;(6) Π k v “ v , for all v P V k with v | Ω z Ω `` k P C p Ω z Ω `` k q ;(7) Π k v P V , and we have ||| Π k v ||| k “ ||| Π k v ||| .Proof. Claims (1)–(3) follow by standard estimates for the Scott-Zhang opera-tor [SZ90, DG12].Assertion (4) is a consequence of the definition (3.7) of Π k since E P G `` k impliesthat N k p E q X G `` k “ N k p E q . Note that v P V p G q implies v | E P P r p E q for all E P G k ONVERGENCE OF
ADGM and thus p Π k v q| E p z q “ v | E p z q for all E P N k p z q if N k p z q X G `` k ‰ H . This is inparticular the case when z P S X N k with S P S `` k .For E P G k z G ` k , we have that N k p z q X G `` k “ H since otherwise there exists E P N k p E q X G `` k and thus E P N k p E q , which implies E P G ` k , thanks to the definitionof G `` k . Therefore, (3.7) implies that Π k v is continuous on Ω z Ω ` k . Moreover, for z P N k p E qX Ω z Ω ` k , definition (3.7) is independent of E and thus Π k v does not jumpacross the boundary Ω z Ω ` k . This completes the proof of (5).On the one hand, if v P V k with v | Ω z Ω ` k P C p Ω z Ω `` k q then we have clearlyΠ k v | Ω z Ω ` k “ v | Ω z Ω ` k . On the other hand, we can conclude Π k v | Ω `` k “ v | Ω `` k from (4).This yields (6).The claim (7) is an immediate consequence of (5). (cid:3) Lemma 18 (Stability) . Let v P V (cid:96) for some k ď (cid:96) P N Yt8u . Then for all E P G k ,we have ˆ E | ∇ Π k v | d x ` ˆ B E h ´ k rr Π k v ss d s À ˆ ω k p E q ˇˇ ∇ pw v ˇˇ d x ` ÿ E P G (cid:96) ,E Ă ω k p E q ˆ B E h ´ (cid:96) rr v ss d s, setting G (cid:96) : “ G ` and h (cid:96) : “ h ` , when (cid:96) “ 8 . In particular, we have ||| Π k v ||| k À ||| v ||| (cid:96) .Proof. We begin by noting that, summing over all elements in G k and accountingfor the finite overlap of the domains ω k p E q , E P G k , the global stability estimate isan immediate consequence of the corresponding local one.We first assume (cid:96) ă 8 . Let E P G `` k Ă G `` (cid:96) . Then, thanks to Lemma 17(4), wehave Π k v | E “ v | E . Moreover, let E P G k such that E X E P S k ; then N k p z q Q E P G `` k and thus p Π k v q| E p z q “ v | E p z q , for all z P N k p E q X N k p E q . Consequently, wehave rr Π k v ss “ rr v ss on B E , in other words ˆ E | ∇ Π k v | d x ` ˆ B E h ´ k rr Π k v ss d s “ ˆ E | ∇ v | d x ` ˆ B E h ´ k rr v ss d s. (3.8)Let now E P G k be arbitrary. Then, an inverse estimate and the local stability(Lemma 17 (1) and (3)) for v E : “ | ω k p E q| ´ ω k p E q v d x P R , imply ˆ E | ∇ Π k v | d x À ˆ E h ´ k | Π k p v ´ v E q| d x À ˆ ω k p E q h ´ k | v ´ v E | d x À ÿ E Ă ω k p E q ,E P G (cid:96) ˆ E | ∇ v | d x ` ˆ B E h ´ (cid:96) rr v ss d s ;(3.9)here the last estimate follows from the broken Poincar´e inequality, Proposition 1.If now for all E P G k , with E Ă ω k p E q , we have E R G `` k , which implies E P G k z G `` k . Then, thanks to Lemma 17(5), we have that Π k v is continuous across B E , i.e., rr Π k v ss | B E “
0. On the contrary, assuming that there exists E P G `` k ,with E P N k p E q , we conclude that E P N k p E q and thus E P G ` . From the localquasi uniformity, we thus have for all E P G (cid:96) with E X E ‰ H that | E | (cid:104) | E | .Let z P N k p E q ; then, according to (3.7), we have that rr Π k v ss | B E p z q “ rr v ss | B E p z q , if D E P N k p z q X G `` k ;0 , else. Using standard scaling arguments, this implies ˆ B E rr Π k v ss d s (cid:104) |B E | ÿ z P N k XB E ` rr Π k v ss | B E p z q ˘ “ |B E | ÿ z P N k XB E ` rr v ss | B E p z q ˘ ď |B E | ÿ z P N (cid:96) XB E ` rr v ss | B E p z q ˘ (cid:104) ˆ B E rr v ss d s. Combining this with (3.9) proves the local bound in the case (cid:96) ă 8 .For (cid:96) “ 8 , we observe that a bound similar to (3.9) can be obtained withLemma 14 instead of Proposition 1. The local bound follows then by arguing as inthe case (cid:96) ă 8 . (cid:3) Corollary 19 (Interpolation estimate) . For v P V (cid:96) , k ď (cid:96) P N Y t8u , we have that ˆ E | ∇ pw v ´ ∇ pw Π k v | d x ` ˆ E h ´ k | v ´ Π k v | ` ˆ B E h ´ k rr v ´ Π k v ss À ˆ ω k p E q | ∇ pw v | d x ` ÿ S P S (cid:96) ,S Ă ω k p E q ˆ S h ´ k rr v ss , where we set G (cid:96) : “ G ` and h (cid:96) : “ h ` , when (cid:96) “ 8 . The constant depends only on d , r and the shape regularity of G .Proof. The claim follows from Lemma 17(3), together with the stability Lemma 18and the local Poincar´e inequality from Proposition 1, respectively, Lemma 14. (cid:3)
The next result concerns the convergence of the quasi-interpolation.
Lemma 20.
Let v P V ; then, ||| v ´ Π k v ||| k Ñ and ||| v ´ Π k v ||| Ñ as k Ñ 8 .Proof.
For brevity, set v k : “ Π k v P V k . Thanks to Lemma 14 and Lemma 17(4)and (5), we have that ||| v ´ v k ||| k À ˆ G k z G `` k | ∇ pw v ´ ∇ pw v k | d x ` ˆ S k z S `` k h ´ k |rr v ´ v k ss| d s ď ˆ G ´ k | ∇ pw v ´ ∇ pw v k | d x ` ˆ S ´ k h ´ k |rr v ´ v k ss| d s “ I ´ k ` II ´ k . We conclude from Lemma 18 that II ´ k “ ˆ S ´ k h ´ k |rr v ´ v k ss| d s À ÿ E P G ´ k ÿ E P G ` ,E Ă ω k p E q ˆ B E h ´ ` rr v ss d s À ˆ S ` z S `` k h ´ ` rr v ss d s. The term on the right hand side is the tail of a convergent series, since it is boundedthanks to ||| v ||| ă 8 and all of its summands are positive. Therefore, II ´ k Ñ k Ñ 8 .Thus, it remains to prove that I ´ k Ñ k Ñ 8 . To this end, we recallthat thanks to the definition of H B Ω XB Ω ´ p Ω ´ q we have that v | Ω ´ “ ˜ v | Ω ´ for somefunction ˜ v P H p Ω q . Since H p Ω q is dense in H p Ω q , for (cid:15) ą
0, there exists v (cid:15) P H p Ω q such that } ˜ v ´ v (cid:15) } H p Ω ´ q ď } ˜ v ´ v (cid:15) } H p Ω q ă (cid:15) . Combining Lemma 17(3) and ONVERGENCE OF
ADGM (1) with standard estimates [SZ90, DG12] for H p Ω q functions, with the Bramble-Hilbert Lemma (see, e.g., [BS02]), we obtain ˆ G ´ k | ∇ pw v ´ ∇ v k | d x À ˆ G ´ k | ∇ v (cid:15) ´ ∇ Π k v (cid:15) | ` | ∇ pw p v ´ v (cid:15) q ´ ∇ Π k p v ´ v (cid:15) q| d x À ˆ N k p G ´ k q h k ÿ | α |“ | D α v (cid:15) | d x ` ˆ N k p G ´ k q | ∇ pw p v ´ v (cid:15) q| d x À } h k χ Ω ´ k } L p Ω q ˆ Ω ÿ | α |“ | D α v (cid:15) | d x ` ˆ N k p G ´ k q | ∇ pw p v ´ v (cid:15) q| d x, where we have used that } h k } L p Ω p N k p G ´ k qqq À } h k χ Ω ´ k } L p Ω q Ñ k Ñ 8 ,thanks to the local quasi-uniformity of G k and Lemma 11. Moreover, we conclude ´ N k p G ´ k q | ∇ pw p v ´ v (cid:15) q| d x Ñ ´ Ω ´ | ∇ p v ´ v (cid:15) q| d x ă (cid:15) from Lemma 11 and the absolutecontinuity of the Lebesgue integral. Consequently, lim k Ñ8 I ´ k À (cid:15) , which completesthe proof of the first claim, since (cid:15) ą S k by S ` and noting that ||| Π k v ||| k “||| Π k v ||| , since Π k v is continuous in Ω z Ω ` . (cid:3) Proof of Lemma 15.
The positivity of |||¨||| on V follows from Lemma 20 togetherwith Corollary 13.In order to prove that V is complete with respect to |||¨||| , let t v (cid:96) u (cid:96) P N Ă V be aCauchy sequence with respect to |||¨||| . Note that thanks to the Friedrichs inequality(Corollary 13), there exists the limit v (cid:96) Ñ v P L p Ω q ; this is the candidate for thelimit of v (cid:96) in V .We first observe that, since v (cid:96) | E P P r for all E P G ` , it follows from the definitionof |||¨||| that v | E P P r for all E P G ` . Moreover, Propositions 4 and 20 imply that t v (cid:96) u (cid:96) P N is also a Cauchy Sequence in BV p Ω q and thus v P BV p Ω q . Therefore, v has L -traces on each B E , E P G k , k P N .Next, we deal with the jump terms. To this end, we first observe that, for k P N , t v (cid:96) u (cid:96) P N is also a Cauchy sequence with respect to the |||¨||| k -norm and uniqueness oflimits imply ““ v (cid:96) ‰‰ | S Ñ rr v ss | S in L p S q as (cid:96) Ñ 8 , for all S P S k , in the sense oftraces. Let (cid:15) ą L “ L p (cid:15) q , such that ˇˇˇˇˇˇ v j ´ v (cid:96) ˇˇˇˇˇˇ k ď ˇˇˇˇˇˇ v j ´ v (cid:96) ˇˇˇˇˇˇ ď (cid:15) for all j ě (cid:96) ě L . Fix (cid:96) ě L , then thanks to Proposition 12, thereexists K ” K p (cid:15), L q , such that for all k ě K , we have ˆ S k z S ` k h ´ ` ““ v L ‰‰ d s ď (cid:15) . Consequently, we have ˆ S k h ´ k rr v ss d s “ ˆ S k z S ` k h ´ k rr v ss d s ` ˆ S ` k h ´ k rr v ss d s “ lim (cid:96) Ñ8 ˆ S k z S ` k h ´ k ““ v (cid:96) ‰‰ d s ` ˆ S ` k h ´ k rr v ss d s. Thus, it follows from ˆ S k z S ` k h ´ k ““ v (cid:96) ‰‰ d s ď ˇˇˇˇˇˇ v (cid:96) ´ v L ˇˇˇˇˇˇ k ` ˆ S k z S ` k h ´ k ““ v L ‰‰ d s ď (cid:15) , (3.10)for (cid:96) ě L , that ˆ S k h ´ k rr v ss d s Ñ ˆ S ` h ´ ` rr v ss d s as k Ñ 8 , (3.11) since (cid:15) ą v | Ω ´ P H B Ω XB Ω ´ p Ω ´ q , i.e., that v is a restriction of a functionfrom H p Ω q to Ω ´ . To this end, for each (cid:96), m P N , we define v (cid:96)m : “ Π m v (cid:96) P V m for (cid:96) ě m P N and since v (cid:96)m P C p Ω z Ω ` k q Ă C p Ω z Ω ` m q (see Lemma 17(5)) for k ě m , wehave that ˇˇˇˇˇˇ v (cid:96)m ˇˇˇˇˇˇ m “ ˇˇˇˇˇˇ v (cid:96)m ˇˇˇˇˇˇ k “ ˇˇˇˇˇˇ v (cid:96)m ˇˇˇˇˇˇ . Thanks to Lemma 20, for each (cid:96) P N , thereexists a monotone sequence t m (cid:96) u (cid:96) P N , such that ˇˇˇˇˇˇ v (cid:96) ´ v (cid:96)m (cid:96) ˇˇˇˇˇˇ ď (cid:96) and thus ˇˇˇˇˇˇ v (cid:96)m (cid:96) ˇˇˇˇˇˇ m (cid:96) “ ˇˇˇˇˇˇ v (cid:96)m (cid:96) ˇˇˇˇˇˇ ď ˇˇˇˇˇˇ v (cid:96) ´ v (cid:96)m (cid:96) ˇˇˇˇˇˇ ` ˇˇˇˇˇˇ v (cid:96) ˇˇˇˇˇˇ ă (cid:96) ` ˇˇˇˇˇˇ v (cid:96) ˇˇˇˇˇˇ ă 8 Consequently, the conforming interpolation I m (cid:96) v (cid:96)m (cid:96) P V m (cid:96) X H p Ω q from Proposi-tion 2 is bounded uniformly in H p Ω q and thus there exists a weak limit ˜ v P H p Ω q of a subsequence, which for convenience we denote with the same label. Moreover,again from Proposition 2, we have ›› v ´ I m (cid:96) v (cid:96)m (cid:96) ›› Ω ´ ď ›› v ´ v (cid:96) ›› Ω ` ›› v (cid:96) ´ v (cid:96)m (cid:96) ›› Ω ` ›› v (cid:96)m (cid:96) ´ I m (cid:96) v (cid:96)m (cid:96) ›› Ω ´ À ›› v ´ v (cid:96) ›› Ω ` (cid:96) ` ˆ S p G m(cid:96) z G ` m(cid:96) q ““ v (cid:96)m (cid:96) ‰‰ d s ď ›› v ´ v (cid:96) ›› Ω ` (cid:96) ` ››› χ Ω ´ m(cid:96) h m (cid:96) ››› L p Ω q ˇˇˇˇˇˇ v (cid:96) ˇˇˇˇˇˇ m (cid:96) , (3.12)which vanishes as (cid:96) Ñ 8 thanks to the Friedrichs inequality (Corollary 13) andLemma 11. Therefore, v | Ω ´ “ ˜ v | Ω ´ and we can define the piecewise gradient of v as in (3.2).We shall next show that ˇˇˇˇˇˇ v ´ v (cid:96) ˇˇˇˇˇˇ Ñ (cid:96) Ñ 8 . Arguing similar as for (3.11),we have ˆ S ` h ´ ` ““ v ´ v (cid:96) ‰‰ d s Ñ (cid:96) Ñ 8 . Consequently, it remains to show that ›› ∇ pw v ´ ∇ pw v (cid:96) ›› Ω Ñ (cid:96) Ñ 8 . To thisend, we observe that t ∇ pw v (cid:96) u (cid:96) is a Cauchy Sequence in L p Ω q d and thus there exists d P L p Ω q d with ›› ∇ pw v (cid:96) ´ d ›› Ω Ñ (cid:96) Ñ 8 and it thus suffices to prove d “ ∇ pw v .Let φ P C p Ω q , then we have from Lemma 20 for the distributional derivative onthe one hand, that x Dv (cid:96) , φ y “ ˆ Ω ∇ pw v (cid:96) ¨ φ d x ´ ˆ S ` ““ v (cid:96) ‰‰ ¨ φ d s Ñ ˆ Ω d ¨ φ d x ´ ˆ S ` rr v ss ¨ φ d s as (cid:96) Ñ 8 . On the other hand, x Dv (cid:96) , φ y “ ˆ Ω z Ω ` k ∇ pw v (cid:96) ¨ φ d x ` ˆ Ω ` k ∇ pw v (cid:96) ¨ φ d x ´ ˆ S ` ““ v (cid:96) ‰‰ ¨ φ d s “ ˆ Ω z Ω ` k ∇I m (cid:96) v (cid:96)m (cid:96) ¨ φ d x ` ˆ Ω z Ω ` k ∇ pw p v (cid:96) ´ I m (cid:96) v (cid:96)m (cid:96) q ¨ φ d x ` ˆ Ω ` k ∇ pw v (cid:96) ¨ φ d x ´ ˆ S ` ““ v (cid:96) ‰‰ ¨ φ d s. In order to estimate the second term, we employ Proposition 2, and obtain for somearbitrary given (cid:15) ą ˆ Ω z Ω ` k ˇˇ ∇ pw p v (cid:96) ´ I m (cid:96) v (cid:96)m (cid:96) q ˇˇ d x À ˆ Ω ˇˇ ∇ pw p v (cid:96) ´ v (cid:96)m (cid:96) q ˇˇ d x ` ˆ Ω z Ω ` k ˇˇ ∇ pw p v (cid:96)m (cid:96) ´ I m (cid:96) v (cid:96)m (cid:96) q ˇˇ d x À (cid:96) ` ˆ S ´ k h ´ m (cid:96) ““ v (cid:96) ‰‰ d s ď (cid:96) ` (cid:15) ADGM for all (cid:96) ě L p (cid:15) q and k ě K p (cid:15), L q similarly as in (3.10). Recalling that ˜ v is the weaklimit of t I m (cid:96) v (cid:96)m (cid:96) u (cid:96) in H p Ω q and v (cid:96) converges strongly in P r p G ` k q , we thus concludethat ˇˇˇˇ ˆ Ω ` χ Ω z Ω ` k ∇ ˜ v ` χ Ω ` k ∇ pw v ´ d ˘ ¨ φ d x ˇˇˇˇ ď (cid:15) } φ } Ω . Recalling (3.2) the assertion follows by letting k Ñ 8 from the uniform integrabilityof ∇ ˜ v and ∇ pw v | Ω ` .Finally note that V k Ă V j for j ě k and thus defining w k : “ v (cid:96)m (cid:96) for k Pt m (cid:96) , . . . , m (cid:96) ` ´ u yields w k P V k . Consequently, } v ´ w k } Ω ` ||| v ´ w k ||| k À ||| v ´ w k ||| “ ˇˇˇˇˇˇ v ´ v (cid:96)m (cid:96) ˇˇˇˇˇˇ ď ˇˇˇˇˇˇ v ´ v (cid:96) ˇˇˇˇˇˇ ` (cid:96) , where we have used that the Friedrichs inequality (Corollary 13) is inherited since ||| v ´ w k ||| “ lim (cid:96) Ñ8 ||| v (cid:96) ´ w k ||| . The right-hand side vanishes because (cid:96) Ñ 8 as k Ñ 8 . (cid:3) Proof of Corollary 16.
The assertion follows from Lemma 15 and the observationthat ||| v ||| À B r v, v s and B r v, w s À ||| v ||| ||| w ||| for all v, w P V . Indeed, the continuity follows with standard techniques using (3.4)and the coercivity is a consequence of ||| Π k v ||| “ ||| Π k v ||| k À B k r Π k v, Π k v s “ B r Π k v, Π k v s and Lemma 20. (cid:3) (Almost) best approximation property In this section we shall prove that the solution u P V of (3.5) is indeed the limitof the discontinuous Galerkin solutions produced by ADGM . This is a consequence ofthe density of spaces t V k u k P N in V and the (almost) best approximation propertyof discontinuous Galerkin solutions; the latter generalises [Gud10]. Lemma 21.
Let u P V be the solution of (3.5) and u k P V k be the DGFEM approximation from (2.6) on G k for some k P N and u the unique solution of thelimit problem from Corollary 16. Then, we have ||| u ´ u k ||| k À ||| u ´ Π k u ||| ` x f, u k ´ Π k u k y Ω ´ B k r Π k u , u k ´ Π k u k s||| u k ´ Π k u ||| k . Proof.
Assume that u k ‰ Π k u P V k X V and set ψ “ u k ´ Π k u . Then, we havefrom (2.7) that α ||| u k ´ Π k u ||| k ď B k r u k ´ Π k u , ψ s “ x f, ψ y Ω ´ B k r Π k u , ψ s“ x f, Π k ψ y Ω ` x f, ψ ´ Π k ψ y Ω ´ B k r Π k u , ψ s“ ` B r u , Π k ψ s ´ B k r Π k u , Π k ψ s ˘ ` ` x f, ψ ´ Π k ψ y Ω ´ B k r Π k u , ψ ´ Π k ψ s ˘ ” p I q ` p II q , using that Π k ψ P V k X V from Lemma 17(7). For p I q , we have, respectively, p I q “ ˆ Ω ∇ pw u ¨ ∇ pw Π k ψ d x ´ ˆ S ` ` tt ∇ u uu ¨ rr Π k ψ ss ` θ tt ∇ Π k ψ uu ¨ rr u ss ˘ d s ` ˆ ˚ S ` ` β ¨ rr u ss rr ∇ Π k ψ ss ` rr ∇ u ss β ¨ rr Π k ψ ss ˘ d s ` ˆ Ω γ ` R prr u ssq ` L p β ¨ rr u ssq ˘ ¨ ` R prr Π k ψ ssq ` L p β ¨ rr Π k ψ ssq ˘ d x ` ˆ S ` σh ` rr u ss ¨ rr Π k ψ ss d s ´ ˆ Ω ∇ pw Π k u ¨ ∇ pw Π k ψ d x ` ˆ S k ` tt ∇ Π k u uu ¨ rr Π k ψ ss ` θ tt ∇ Π k ψ uu ¨ rr Π k u ss ˘ d s ´ ˆ ˚ S ` ` β ¨ rr Π k u ss rr ∇ Π k ψ ss ` rr ∇ Π k u ss β ¨ rr Π k ψ ss ˘ d s ´ ˆ Ω γ ` R k prr Π k u ssq ` L k p β ¨ rr Π k u ssq ˘ ¨ ` R k prr Π k ψ ssq ` L k p β ¨ rr Π k ψ ssq ˘ d x ´ ˆ S ` σh k rr Π k u ss ¨ rr Π k ψ ss d s “ ˆ Ω ∇ pw p u ´ Π k u q ¨ ∇ pw Π k ψ d x ´ ˆ S ` k tt ∇ p u ´ Π k u quu ¨ rr Π k ψ ss d s ´ θ ˆ S ` tt ∇ Π k ψ uu ¨ rr u ´ Π k u ss d s ` ˆ ˚ S ` ` β ¨ rr u ´ Π k u ss rr ∇ Π k ψ ss ` rr ∇ u ´ ∇ Π k u ss β ¨ rr Π k ψ ss ˘ d s ` ˆ Ω γ ` R prr u ´ Π k u ssq ` L p β ¨ rr u ´ Π k u ssq ˘ ¨ ` R prr Π k ψ ssq ` L p β ¨ rr Π k ψ ssq ˘ d x ` ˆ S ` k σh k rr u ´ Π k u ss ¨ rr Π k ψ ss d s À ||| u ´ Π k u ||| ||| Π k ψ ||| “ ||| u ´ Π k u ||| ||| Π k ψ ||| k À ||| u ´ Π k u ||| ||| u k ´ Π k u ||| k ;here we used that Π k u , Π k ψ P V k X V , h “ h k on S ` k and that Π k u andΠ k ψ are continuous on Ω z Ω ` k , i.e., rr Π k u ss “ rr Π k ψ ss “ S ` z S ` k , which followsfrom Lemma 17. Note that this and rr Π k u ss “ rr Π k ψ ss “ Bp Ω z Ω ` k q fromLemma 17 also implies that L k p Π k ψ q “ L p Π k ψ q and L k p Π k u q “ L p Π k u q as well as the corresponding relations between R k and R ; compare with (3.3).Thus, the above estimate follows from the Cauchy-Schwarz inequality, applicationof inverse inequalities in conjunction with the stability of the lifting operators (3.4),and Lemma 18.Consequently, triangle inequality and the above imply ||| u ´ u k ||| k ď ||| u ´ Π k u ||| k ` ||| u k ´ Π k u ||| k À ||| u ´ Π k u ||| k ` ||| u ´ Π k u ||| ` x f, ψ ´ Π k ψ y Ω ´ B k r Π k u , ψ ´ Π k ψ s||| u k ´ Π k u ||| k . Thanks to ||| u ´ Π k u ||| k ď ||| u ´ Π k u ||| , this proves the assertion. (cid:3) The properties of the quasi-interpolation (3.6) allow for the consistency term inLemma 21 to be bounded by the a posteriori indicators of essentially the elements,which will experience further refinements.
ONVERGENCE OF
ADGM Lemma 22.
Let u P V be the solution of (3.5) and u k P V k be the DGFEM approximation from (2.6) on G k for some k P N . Then, we have x f, u k ´ Π k u k y Ω ´ B k r Π k u , u k ´ Π k u k s||| u k ´ Π k u ||| k À ´ ÿ E P G k z G ` k E k p Π k u , E q ¯ { , where G ` k : “ t E P G k : N k p E q Ă G `` k u .Proof. Let v k : “ Π k u and φ : “ u k ´ Π k u k “ u k ´ Π k u ´ Π k p u k ´ Π k u q . Then,using integration by parts, we have x f, φ y Ω ´ B k r v k , φ s“ ˆ G k p f ` ∆ v k q φ d x ´ ˆ S k rr ∇ v k ss tt φ uu d s ` ˆ S k θ tt ∇ φ uu rr v k ss d s ´ ˆ ˚ S k ` β ¨ rr v k ss rr ∇ φ ss ` rr ∇ v k ss β ¨ rr φ ss ˘ d s ´ ˆ Ω γ ` R k prr v k ssq ` L k p β ¨ rr v k ssq ˘ ¨ ` R k prr φ ssq ` L k p β ¨ rr φ ssq ˘ d x ´ σ ˆ S k h ´ k rr v k ss rr φ ss d s. Thanks to properties of Π k (see Lemma 17), we have that rr v k ss | S ” S P S k z S ` k , rr v k ss | Ω z Ω ` k ” φ | E ” E P G `` k , and rr φ ss | S ” S P S `` k . Therefore, wehave x f, φ y Ω ´ B k r v k , φ s“ ˆ G k z G `` k p f ` ∆ v k q φ d x ´ ˆ S k z S `` k rr ∇ v k ss tt φ uu d s ` θ ˆ S ` k tt ∇ φ uu rr v k ss d s ´ ˆ ˚ S ` k β ¨ rr v k ss rr ∇ φ ss d s ´ ˆ ˚ S k z S `` k rr ∇ v k ss β ¨ rr φ ss d s ´ ˆ Ω γ ` R k prr v k ssq ` L k p β ¨ rr v k ssq ˘ ¨ ` R k prr φ ssq ` L k p β ¨ rr φ ssq ˘ d x ´ σ ˆ S ` k z S `` k h ´ k rr v k ss rr φ ss d s . (4.1)The last term on the right-hand side of (4.1) can be estimated using Cauchy-Schwarz’ inequality; for the first two terms we use the interpolation estimates fromCorollary 19 for φ “ ψ ´ Π k ψ with ψ “ u k ´ Π k u P V k as to obtain ˆ G k z G `` k p f ` ∆ v k q φ d x ´ ˆ S k z S `` k rr ∇ v k ss tt φ uu d s À «´ ˆ G k z G `` k h k | f ` ∆ v k | d x ¯ { ` ´ ˆ S k z S `` k h k rr ∇ v k ss d s ¯ { ff ||| u k ´ Π k u ||| k . Moreover, from φ | E ” E P G `` k , we have that φ | ω k p S q ” tt ∇ φ uu| S ” S P S ` k “ S p G ` k q . Therefore, by standard trace inequalities, inverse estimatesand Corollary 19, we have that ˆ S ` k tt ∇ φ uu rr v k ss d s “ ˆ S ` k z S ` k tt ∇ φ uu rr v k ss d s À ´ ˆ S ` k z S ` k h ´ k rr v k ss d s ¯ { ||| φ ||| k . A similar argument yields ˆ ˚ S ` k β ¨ rr v k ss rr ∇ φ ss d s “ ˆ ˚ S ` k z S ` k β ¨ rr v k ss rr ∇ φ ss d s À | β | ´ ˆ ˚ S ` k z S ` k h ´ k rr v k ss d s ¯ { ||| φ ||| k . Finally we have with (2.4c) and the local support of the local liftings, that ˆ Ω R k prr v k ssq ¨ R k prr φ ssq d x “ ˆ Ω ` ÿ S P S ` k R Sk prr v k ssq ˘ ¨ ` ÿ S P S k z S `` k R Sk prr φ ssq ˘ d x “ ˆ G ` k z G `` k R k prr v k ssq ¨ R k prr φ ssq d x À ` ˆ S ` k z S ` k h ´ k rr v k ss d s ¯ { ||| φ ||| k . Similar bounds hold for the remaining terms in (4.1). Combining the above obser-vations proves the desired assertion. (cid:3)
In order to conclude convergence of the sequence of discrete discontinuous Galerkinapproximations from Lemma 22, we need to control the error estimator. To thisend, we shall use Verf¨urth’s bubble function technique.Let n P N , such that n uniform refinements of an element ensure that the elementas well as each of its sides have at least one interior node. We specify the elementsin G k , which neighbourhood is eventually uniformly refined n times by G k : “ (cid:32) E P G k : D (cid:96) “ (cid:96) p E q ě k ` n such thatall E P N k p E q are n times uniformly refined in G (cid:96) ( This guarantees that suitable discrete interior and side bubble functions are availablein V for all E P G k ; (compare also with [D¨or96], [MNS00] and [MSV08]). We defineΩ k : “ Ω p G k q Ă Ω ´ k . Introducing Ω ‹ k “ Ω p G ‹ k q with G ‹ k : “ G k zp G `` k Y G k q , we havefrom [MSV08, (4.15)] that | Ω p G ‹ k q| Ñ k Ñ 8 . (4.2) Proposition 23.
Let u be the solution of (3.5) . Then, for every E P G k and v P V k , k P N , we have ˆ E h k | f ` ∆ v | d x ` ˆ B E X Ω h k ““ ∇ pw v ‰‰ d s À ›› ∇ pw p u ´ v q ›› ω k p E q ` ˆ t S P S ` : S Ă ω k p E qu h ´ ` rr u ´ v ss d s ` osc p N k p E q , f q ; in particular, we also have ÿ E P G k ˆ E h k | f ` ∆ v | d x ` ˆ B E X Ω h k ““ ∇ pw v ‰‰ d s À ||| u ´ v ||| ` ÿ E P G k ÿ E P ω k p E q osc p E , f q . Note that since v P V k Ć V in general, the above terms may be equal to infinity.Proof. The proof follows from standard techniques; compare e.g. [KP03, BN10] byreplacing Verf¨urth’s bubble functions by their discrete counterparts. However, inorder to keep the presentation self-contained, we provide a sketch of the proof. Let E P G k , then, thanks to the definition of G k , there exists some (cid:96) ą k such that ONVERGENCE OF
ADGM there exists a piecewise affine discrete bubble function φ E P V (cid:96) X C p Ω q satisfying φ E P H p E q , and h dE } ∇ qφ E } L p E q À } ∇ qφ E } E À h ´ E } q } E for all q P P r ´ p E q ;(4.3)compare [D¨or96, MSV08]. Let f E P P r ´ p E q an arbitrary polynomial. Observingthat p f E ` ∆ v q φ E P C p Ω q and thus does not jump across sides, we have by equiva-lence of norms on finite dimensional spaces and a scaled trace inequality, that ˆ E | f E ` ∆ v | d x À ˆ E p f E ` ∆ v qp f E ` ∆ v q φ E d x “ B r u ´ v, p f E ` ∆ v q φ E s ´ ˆ E p f ´ f E qp f E ` ∆ v q φ E d x À ›› ∇ pw p u ´ v q ›› E } ∇ p f E ` ∆ v q φ E } E ´ ˆ S ` rr u ´ v ss tt ∇ p f E ` ∆ v q φ E uu d s ` } f ´ f E } E }p f E ` ∆ v q φ E } E . From (4.3) and standard inverse estimates, we conclude that ˇˇˇˇ ˆ S ` rr u ´ v ss tt ∇ p f E ` ∆ v q φ E uu d s ˇˇˇˇ ď ÿ S P S ` ,S Ă E ˆ S rr u ´ v ss d s } ∇ p f E ` ∆ v q φ E } L p E q À ´ ˆ S ` h d ´ ` rr u ´ v ss d s ¯ { h ´ ´ d E } f E ` ∆ v } E À ´ ˆ S ` h ´ ` rr u ´ v ss d s ¯ { h ´ E } f E ` ∆ v } E , since h ` ď h E on E . Therefore, we arrive at ˆ E h k | f E ` ∆ v | d x À ›› ∇ pw p u ´ v q ›› E ` ÿ S P S ` ,S Ă E ˆ S h ´ ` rr u ´ v ss d s ` h E } f ´ f E } E . (4.4)Thanks to the definition of G k , the same bound applies for all E P N k p E q .We now turn to investigate the jump terms. To this end, we fix one S P ˚ S k , S Ă E . Thanks to the definition of G k , there exists a hat function φ S P V (cid:96) X C p Ω q X H p ω k p S qq , and for q P P r ´ p S q , let ˜ q P P r ´ p ω k p S qq be some extension, such that h dE } ∇ ˜ qφ S } L p ω k p S qq À } ˜ qφ S } ω k p S q À h E ˆ S | q | d s. (4.5) Noting that rr ∇ v ss P P r ´ p S q , we have, by the equivalence of norms on finite dimen-sional spaces, that ˆ S rr ∇ v ss d s À ˆ S rr ∇ v ss φ S d s “ B r u ´ v, Č rr ∇ v ss φ S s ´ ˆ ω k p S q p f ` ∆ v q Č rr ∇ v ss φ S d x À ›› ∇ pw p u ´ v q ›› ω k p S q ››› ∇ Č rr ∇ v ss φ S ››› ω k p S q ` ˆ S ` rr u ´ v ss tt ∇ Č rr ∇ v ss φ S uu d s ` ` } f ` ∆ v } E ` } f ` ∆ v } E ˘ ›››Č rr ∇ v ss φ S ››› ω k p S q . Similarly, as for the element residual, we have that ˆ S ` rr u ´ v ss tt ∇ Č rr ∇ v ss φ S uu d s À ´ ÿ S P S ` ,S Ă ω k p S q h ´ ` rr u ´ v ss ¯ ´ ˆ S h E rr ∇ v ss d s ¯ , using (4.5). Combining this with (4.5), we obtain ˆ S h E rr ∇ v ss d s À ›› ∇ pw p u ´ v q ›› ω k p S q ` ÿ S P S ` ,S Ă ω k p S q ˆ S h ´ ` rr u ´ v ss d s ` h E } f ` ∆ v } E ` h E } f ` ∆ v } E . Finally applying the bound (4.4) to
E, E P N k p E q , we have proved the first asser-tion.The second assertion follows, then, by summing over all E P G k together withan observation from [MSV08], which we sketch here in order to keep the this workself-contained. Let M : “ max t N k p E q : E P G k u be the maximal number ofneighbours, then G k can be split into M ` G k, , . . . , G k,M such that foreach j , we have that E , E P G k,j with E ‰ E implies that N k p E q X N k p E q “ H .Consequently, we have ÿ E P G k ›› ∇ pw p u ´ v q ›› ω k p E q ď M ÿ j “ ÿ E P G k,j ›› ∇ pw p u ´ v q ›› ω k p E q ď p M ` q ›› ∇ pw p u ´ v q ›› k . Together with similar estimates for the jump terms and the oscillations the secondassertion follows from the first one. (cid:3)
Theorem 24.
Let u the solution of (3.5) and u k P V k be the DGFEM approxi-mation from (2.6) on G k for some k P N . Then, ||| u ´ u k ||| k Ñ as k Ñ 8 and in particular } u ´ u k } Ω Ñ as k Ñ 8 .Proof.
Thanks to Lemma 21, Lemma 20 and Lemma 22, we have thatlim k Ñ8 ||| u ´ u k ||| k À lim k Ñ8 ||| u ´ v k ||| ` ÿ E P G k z G ` k E k p v k , E q “ lim k Ñ8 ÿ E P G k z G ` k E k p v k , E q , ONVERGENCE OF
ADGM where v k : “ Π k u . We conclude from (4.2) that ˇˇ Ω z ` Ω k Y Ω ` k ˘ˇˇ ď ˇˇ Ω zp Ω k Y Ω `` k q ˇˇ ` | Ω `` k z Ω ` k | Ñ , as k Ñ 8 . Indeed, for k P N , it follows from Lemma 10 and G ` k ă 8 , that thereexists K “ K p k q , such that G ` k Ă G ` K , i.e. | Ω ` z Ω ` K | ď | Ω ` z Ω ` k | Ñ k Ñ 8 .Thanks to monotonicity we conclude that | Ω `` k z Ω ` k | ď | Ω ` z Ω ` k | Ñ k Ñ 8 .We next show that this implies ÿ E P G k zp G k Y G ` k q E k p v k , E q Ñ . Lemma 20 implies that ||| u ´ v k ||| Ñ ˆ S p G k zp G k Y G ` k qq h ´ k rr v k ss d s À ˆ S p G k z G ` k q h ´ k rr u ss d s ` ||| u ´ v k ||| k ď ˆ S p G ` z G ` k q h ´ ` rr u ss d s ` ||| u ´ v k ||| k . The last term on the right-hand side of the above estimate vanishes thanks toLemma 20. Again, letting K “ K p k q , such that G ` k Ă G ` K , we have ˆ S p G ` z G ` K p k q q h ´ ` rr u ss d s ď ˆ S p G ` z G ` k q h ´ ` rr u ss d s Ñ , as k Ñ 8 . Thanks to monotonicity, we thus conclude ´ S p G ` z G ` k q h ´ ` rr u ss d s Ñ
0, as k Ñ 8 .On the remaining elements G ´ k , it follows from Proposition 23 that ÿ E P G k E k p v k , E q À ||| u ´ v k ||| ` ÿ E P G k osc p N k p E q , f q . The first term on the right-hand side vanishes due to Lemma 20. For the second termwe observe that | Ť t ω k p E q : E P G k u| À | Ω k | , depending on the shape regularity of G and, therefore, it vanishes since ››› h k χ Ω k ››› L p Ω q ď ››› h k χ Ω ´ k ››› L p Ω q Ñ k Ñ 8 , (4.6)thanks to Lemma 11.The second limit follows then from } u ´ u k } Ω ď } u ´ Π k u } Ω ` } Π k u ´ u k } Ω À ||| u ´ Π k u ||| ` ||| Π k u ´ u k ||| k , which vanishes due to the above observations. (cid:3) Proof of the main result
We are now in the position to prove that the error estimator vanishes by splittingthe estimator according to G k “ G k Y G `` k Y G ‹ k (5.1)and consider each part separately following the ideas of [MSV08]. This in turnimplies that the sequence of discontinuous Galerkin approximations produced by ADGM indeed converges to the exact solution of (2.1).
Lemma 25.
We have that E k p G k q Ñ , as k Ñ 8 . Proof.
Thanks to Proposition 23, we have ÿ E P G k ˆ E h k | f ` ∆ u k | d x ` ˆ B E X Ω h k rr ∇ u k ss d s À ||| u ´ u k ||| ` ÿ E P G k osc p N k p E q , f q . The right-hand side vanishes thanks to Theorem 24 and (4.6).It remains to prove that ˆ S p G k q h ´ k rr u k ss d s Ñ , as k Ñ 8 . By definition, Ω k Ă Ω z Ω ` k and, thanks to Lemma 17(5), we have that Π k u P C p Ω z Ω ` k q . Therefore, we conclude ˆ S p G k q h ´ k rr u k ss d s “ ˆ S p G k q h ´ k rr u k ´ Π k u ss d s ď ||| u k ´ Π k u ||| k Ñ k Ñ 8 ; see Lemma 20 and Theorem 24. (cid:3)
Lemma 26.
We have that lim k Ñ8 E k p G ‹ k q “ . Proof.
We conclude from the lower bound (Proposition 6) that ÿ E P G ‹ k ˆ E h k | f ` ∆ u k | d x ` ˆ B E h k rr ∇ u k ss d s À ÿ E P G ‹ k } u ´ u k } ω k p E q ` ›› ∇ u ´ ∇ pw u k ›› ω k p E q ` osc p N k p E q , f q À ÿ E P G ‹ k ! } u } ω k p E q ` } u ´ u k } ω k p E q ` } u } ω k p E q ` } ∇ u } ω k p E q ` ›› ∇ pw u ´ ∇ pw u k ›› ω k p E q ` ›› ∇ pw u ›› ω k p E q ` osc p N k p E q , f q ) . This vanishes as k Ñ 8 thanks to Theorem 24 and (4.2), together with the uniformintegrability of the terms involving u and u . Note that ˇˇ Ť t ω k p E q : E P G ‹ k u ˇˇ À| Ω ‹ k | , with the constant depending on the shape regularity of G .It remains to prove ˆ S p G ‹ k q h ´ k rr u k ss d s Ñ , as k Ñ 8 . To this end, we observe that ˆ S p G ‹ k q h ´ k rr u k ss d s “ ˆ S p G ‹ k q h ´ k rr u k ´ Π k u ss d s ` ˆ S p G ‹ k q h ´ k rr Π k u ss d s ď σ ||| u k ´ Π k u ||| k ` ˆ S p G ‹ k q h ´ k rr Π k u ss d s. As in the proof of Lemma 25, we have that the first term vanishes as k Ñ 8 .Thanks to Lemma 10, there exists (cid:96) p k q ě K p k q ě k such that G ` k Ă G `` K p k q and G ` K p k q Ă G `` (cid:96) p k q . Consequently, we have that rr Π (cid:96) u ss | S “ S P G k ; see ONVERGENCE OF
ADGM Lemma 17(5). Therefore, we conclude from Lemma 20 that σ ˆ S p G ‹ k q h ´ k rr Π k u ss d s “ σ ˆ S p G ‹ k q h ´ k rr Π k u ´ Π (cid:96) u ss d s À ||| Π k u ´ u ||| k ` ||| u ´ Π (cid:96) u ||| (cid:96) Ñ , as k Ñ 8 . (cid:3) Lemma 27.
We have E k p G `` k q Ñ as k Ñ 8 . Proof.
Step 1:
By definition, elements in G `` k will not be subdivided, i.e. we havethat M k Ă G k z G `` k ; compare with (2.9). As a consequence of Lemmas 25 and 26,we conclude from (2.8) for all E P G `` k that E k p E q ď lim k Ñ8 g p E k p M k qq “ lim k Ñ8 g p E k p G ´ k Y G ‹ k qq Ñ , (5.2)as k Ñ 8 . We shall reformulate the above element-wise convergence in an integralframework, in order to conclude E k p G `` k q Ñ k Ñ 8 via a generalised version ofthe dominated convergence theorem. To this end, we shall consider some propertiesof the error indicators.
Step 2:
Thanks to the definition of G `` k , we have for all E P G `` k , that ω k p E q “ ω (cid:96) p E q “ : ω p E q and N k p E q “ N (cid:96) p E q “ N p E q for all (cid:96) ě k . Therefore, we obtain bythe lower bound, Proposition 6, that E k p E q À ||| u k ´ u ||| N p E q ` osc p N p E q , f q À ||| u k ´ u ||| N p E q ` } u } N p E q ` } u } H p ω p E qq ` } f } ω p E q “ : ||| u k ´ u ||| N p E q ` C E . (5.3)Arguing as in the proof of Proposition 23, we can conclude from the local estimatethat ÿ E P G `` k C E À ||| u ||| ` } u } H p Ω q ` } f } ă 8 (5.4)independently of k . Step 3:
We shall now reformulate E k p G `` k q in integral form. Note that thanks toLemma 10, we have that G ` “ Ť k P N G ` k “ Ť k P N G `` k , and also that the sequence t G `` k u k P N is nested. For x P Ω ` , let (cid:96) “ (cid:96) p x q : “ min t k P N : there exists E P G `` k such that x P E u . Then, we define (cid:15) k p x q : “ M k p x q : “ k ă (cid:96), and (cid:15) k p x q : “ | E | E k p E q , M k : “ | E | ´ ||| u k ´ u ||| N p E q ` C E ¯ for k ě (cid:96). Consequently, for any k P N , we have E k p G `` k q “ ˆ Ω ` (cid:15) k p x q d x. Moreover, thanks to the fact that the sequence t G `` k u k P N is nested, we concludefrom (5.2) that lim k Ñ8 (cid:15) k p x q “ lim k Ñ8 | E | E k p E q “ . It follows from (5.3) and (5.4) that M k is an integrable majorant for (cid:15) k . Step 4:
We shall show that the majorants t M k u k P N converge in L p Ω ` q to M p x q : “ | E | C E , for x P E and E P G ` . Then the assertion follows from a generalised majorised convergence theorem; see[Zei90, Appendix (19a)]. In fact, by the definition of M k , we have that } M k ´ M } L p Ω ` q “ ÿ E P G `` k } M k ´ M } L p E q ` ÿ E P G ` z G `` k } M } L p E q . The latter term vanishes since it is the tail of a converging series (compare with (5.4))and for the former term, we have, thanks to Theorem 24, that ÿ E P G `` k } M k ´ M } L p E q “ ÿ E P G `` k ||| u k ´ u ||| N p E q À ||| u k ´ u ||| k Ñ k Ñ 8 . (cid:3) Proof of Theorem 9.
The assertion follows from Proposition 5 together with Lem-mas 25, 26, and 27 recalling the splitting (5.1). (cid:3)
Acknowledgement
We thank the anonymous referee of the paper [DGK19] for finding a highly non-trivial counterexample to the first statement in [DGK19, Lemma 11], which lead tothis corrected version of [KG18].
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URL : E-mail address : [email protected] Emmanuil H. Georgoulis, Department of Mathematics, University of Leicester, Uni-versity Road, Leicester, LE1 7RH, United Kingdom and Department of Mathematics,School of Applied Mathematical and Physical Sciences, National Technical Universityof Athens, Zografou 157 80, Greece
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