c 2017 Society for Industrial and Applied Mathematics

Transcription

1 SIAM J. NUMER. ANAL. Vol. 55, No. 4, pp c 017 Socity for Industrial and Applid Mathmatics ON HANGING NODE CONSTRAINTS FOR NONCONFORMING FINITE ELEMENTS USING THE DOUGLAS SANTOS SHEEN YE ELEMENT AS AN EXAMPLE WOLFGANG BANGERTH, IMBUNM KIM, DONGWOO SHEEN, AND JAERYUN YIM Abstract. On adaptivly rfind quadrilatral or hxahdral mshs, on usually mploys constraints on dgrs of frdom to dal with hanging nods. How ths constraints ar constructd is rlativly straightforward for conforming finit lmnt mthods: Th constraints ar usd to nsur that th discrt solution spac rmains a subspac of th continuous spac. On th othr hand, for nonconforming mthods, this guiding principl is not availabl and on nds othr ways of nsuring that th discrt spac has dsirabl proprtis. In this papr, w invstigat how on would construct hanging nod constraints for nonconforming lmnts, using th Douglas Santos Shn Y (DSSY) lmnt as a prototypical cas. W idntify thr possibl stratgis, two of which lad to provably convrgnt schms with diffrnt proprtis. For both of ths, w show that th structur of th constraints diffrs qualitativly from th way constraints ar usually dalt with in th conforming cas. Ky words. finit lmnt mthod, adaptiv mshs, hanging nods, constraints AMS subjct classifications. 65N30, 65N16, 65N50 DOI /16M Introduction. Rfining individual clls of a finit lmnt msh for xampl, in adaptiv msh rfinmnt schms basd on rror stimators crats facs with so-calld hanging nods, i.., vrtics that ar locatd in th intrior of dgs or facs of som of th surrounding lmnts. On such facs, shap functions cannot asily b dfind without losing som of th proprtis of th finit lmnt spac that ar ncssary to show convrgnc at th appropriat ordr. Howvr, thr ar wll-known stratgis for rsolving this problm, ithr through th us of transition lmnts that ntirly rmov th hanging nods (for xampl, in th rdgrn rfinmnt stratgy for triangls; s [5]), or through th us of constraints on dgrs of frdom that nsur that th critical proprty of th finit lmnt ncssary for th thory is rstord [16, 5, 19, 0]. Whil both of ths approachs can asily b usd for triangular or ttrahdral mshs, a lack of practical and widly Rcivd by th ditors April 19, 016; accptd for publication (in rvisd form) Fbruary 16, 017; publishd lctronically July 13, Funding: Th work of th first author was supportd by th National Scinc Foundation through award OCI and by th Computational Infrastructur in Godynamics (CIG) initiativ, through th National Scinc Foundation undr award EAR , and th Univrsity of California Davis. Th work of th first and third author s was supportd in part by National Rsarch Foundation of Kora NRF-014R1AA1A Th work of th third author was also supportd by th Nxt-Gnration Information Computing Dvlopmnt Program through th National Rsarch Foundation of Kora (NRF) fundd by th Ministry of Education, Scinc and Tchnology (NRF-015M3C4A706566). Dpartmnt of Mathmatics, Colorado Stat Univrsity, Fort Collins, CO 8053 (bangrth@colostat.du). Dpartmnt of Mathmatics, Soul National Univrsity, Soul 0886, Rpublic of Kora, and Intrdisciplinary Program in Computational Scinc and Tchnology, Soul National Univrsity, Soul 0886, Rpublic of Kora (ibkim11@gmail.com, dongwooshn@gmail.com). Intrdisciplinary Program in Computational Scinc and Tchnology, Soul National Univrsity, Soul 0886, Rpublic of Kora (jaryun.yim@gmail.com). 1719

2 170 W. BANGERTH, I. KIM, D. SHEEN, AND J. YIM usd transition lmnt stratgis rquirs th us of constraints for quadrilatral and hxahdral mshs. Whn using conforming finit lmnts, th construction of ths constraints is typically straightforward: w us th constraints to rstor th conformity proprty for th finit lmnt spac. For xampl, for H 1 -conforming lmnts, on nds to rstor th continuity of th finit lmnt spac along th affctd fac with a hanging nod. In th cas of conforming bilinar lmnts, this implis th familiar constraint that a dgr of frdom U H dfind on a hanging nod nds to satisfy U H = 1 U L + 1 U R, whr U L, U R ar th dgrs of frdom dfind at th nds of th parnt dg on which th hanging nod livs. Similar constructions ar usd for H(curl)- and H(div)-conforming lmnts whr constraints ar usd to nforc th continuity of tangntial or normal componnts of th finit lmnt fild, rspctivly. On th othr hand, th situation is not so clar for nonconforming lmnts. Thr, th discrt finit lmnt spac is alrady nonconforming with rgard to th continuous solution spac, vn on mshs without hanging nods. Consquntly, thr is no point in rstoring conformity along facs with hanging nods, and conformity will not b usful as th guiding principl for construction of constraints. At th sam tim, nonconforming lmnts sitting somwhr btwn conforming and ntirly discontinuous lmnts hav nough intrnal structur that thy do not nd th addition of pnalty trms to th bilinar form, as is ncssary, for xampl, for th discontinuous Galrkin (DG) mthod. Consquntly, th guiding principl should b to com up with som kind of constraint at hanging nods that rstors th critical proprty of th discrt spac that allows us to avoid pnalty trms. In this contribution, w invstigat this situation for th nonconforming Douglas Santos Shn Y (DSSY) lmnt [1] that can b usd in th discrtization of th Laplac and rlatd quations. Th DSSY lmnt provids optimal convrgnc ordr, but bcaus its dgrs of frdom ar dfind on th facs of clls rathr than thir vrtics, it has a smallr stncil and consquntly sparsr systm matrix than th rgular, H 1 -conforming Lagrang lmnts. Basd on analogis with othr lmnts and knowldg of th proof of convrgnc of th DSSY lmnt, w will introduc thr possibl ways to construct constraints at hanging nods. Whil th most obvious choic (latr calld option A ) turns out to not b convrgnt, th othr two approachs ar in fact convrgnt. On of ths ( option B ) is built in such a way that it rstors th critical ingrdint in th original proof of optimal convrgnc found in [1]. Howvr, this construction of constraints dos not just coupl dgrs of frdom locatd on th fac with th hanging nod (as is th cas for all conforming lmnts w ar awar of) but indd all dgrs of frdom on th ntir largr cll. Bcaus ths may in turn b constraind thmslvs, th stncil of constraind dgrs of frdom may in fact xtnd far byond a cll and its immdiat nighbors. This has significant implications for implmntations that w will discuss as wll. To addrss this problm, w invstigat a wakr, altrnativ choic ( option C ) of constraints that dos not shar this drawback but rquirs mor work to prov convrgnc. Th rmaindr of this papr is structurd as follows. Sction provids an ovrviw of th DSSY lmnt, and sction 3 introducs furthr notation ncssary in th discussions that follow. Sction 4 thn tracs out th proof of convrgnc on mshs without hanging nods and in particular points out th critical pics that gt lost if a msh dos hav hanging nods; it also prsnts thr possibl constructions of constraints that could rasonably b usd at hanging nods. Sctions 5 and 6 thn covr th analysis of th two altrnativ constructions that ar in fact convrgnt,

3 NONCONFORMING FINITE ELEMENTS WITH HANGING NODES 171 followd by numrical xampls in sction 7 that quantitativly assss th options prviously idntifid. W conclud in sction 8. Rmark 1.1. Whil our discussions hrin ar basd spcifically on th DSSY lmnt, w intnd th construction of constraints for this lmnt to also provid guidanc for othr nonconforming lmnts, such as th simplicial Crouzix Raviart lmnt [9] or th quadrilatral Rannachr Turk lmnt [15], for which w bliv that similar principls will apply. Our discussions blow will b rstrictd to two dimnsions bcaus th dgr of notation, as wll as th graphics ncssary to build intuition, would b too cumbrsom without providing additional insight. Howvr, th xtnsion of our work to thr dimnsions is obvious. Rmark 1.. Nithr nonconforming lmnts nor adaptivly rfind mshs using hanging nods ar nw. As a consqunc, it is prhaps surprising that w hav not bn abl to find much litratur on thir combination. Carstnsn and Hu [6] mak an assumption that is quivalnt to option C discussd in sction 4. but do not discuss altrnativs, or laborat on practical implications of this assumption. Huang and Xi [13] also us a similar assumption, but in th contxt of avoiding hanging nod constraints whn daling with conforming lmnts at hanging nods. Finally, on of th rviwrs of this papr pointd out th Ph.D. thsis of Schmid [17]. Th thsis was publishd as a book (in Grman), is now out of print, and is consquntly difficult to accss. W hav not bn abl to vrify that it contains matrial rlatd to our work, though w do not doubt it. An arlir papr by Schmid and Wagnr [18, sction 4.3] has a rmark that stats that nods on hanging dgs nd to b tratd diffrntly from thos on rgular dgs a statmnt that at th tim may not hav bn as obvious as it appars today; it dos not, howvr, provid dtails on how spcifically this tratmnt should look, nor any analysis.. A brif ovrviw of th DSSY lmnt. Th DSSY lmnt [1] was dvlopd for th discrtization of scond-ordr lliptic problms such as th Poisson quation: (.1a) (.1b) u = f in Ω, u = g on Ω. For ths quations, th appropriat solution spac is V = H 1 (Ω). As a typical nonconforming finit lmnt discrtization, th DSSY lmnt dfins a finit dimnsional spac V h that dos not satisfy th usual condition V h V. Rathr, it contains functions that ar discontinuous and consquntly not mmbrs of V. Mor spcifically, th DSSY lmnt is dfind as follows [4, 1]. Lt K = [ 1, 1] b th rfrnc lmnt and T = {K} a subdivision of Ω into quadrilatrals that ar mappd from th rfrnc cll by bilinar mappings φ K. Th DSSY spac V is thn dfind on th rfrnc cll as (.) V = span{1, x 1, x, θ( x 1 ) θ( x ), x 1 x }, whr θ can b dfind as ithr θ(ξ) = ξ 5 3 ξ4 or θ(ξ) = ξ 5 6 ξ4 + 7 ξ6. Th choic of ths options will b immatrial to th rmaindr of this papr, but th additional quartic or sxtic trm nabls th finit lmnt spac V to fulfill th man

4 17 W. BANGERTH, I. KIM, D. SHEEN, AND J. YIM valu proprty 1 (.3) v( mê) = v ê for all dgs ê of th rfrnc cll K; mê dnots th cntr point of dg ê. Hr, and in what follows, s f = 1 s f dnots th avrag Lbsgu intgral of f ovr s s. Th addition of th last shap function in th dfinition (.) (which lads to an intrior bubbl function whn choosing a st of dual functionals for V ) is ncssary to guarant th optimal convrgnc ordr on gnral quadrilatral mshs if ths shap functions ar mappd from a standard rfrnc cll (s [4, 1]); it can b omittd on mshs consisting only of paralllograms. This spac cannot b connctd in a continuous way across clls to form a global spac sufficintly rich to approximat th solutions of partial diffrntial quations. Rathr, th DSSY construction only rquirs that th man valus of functions ar qual on intrfacs γ btwn nighboring clls K(γ) + and K(γ). On can quivalntly rquir continuity at dg midpoints, but using th viwpoint of quality of man valus will turn out to yild asir-to-undrstand insights during th proofs blow; w will thrfor bas our argumnts on this prspctiv. If w dfin E i to b th st that consists of all intrior dgs of th msh, and assuming w hav a msh without hanging nods, thn w can construct th global spac as follows: { } V h (T) = u h L (.4) (Ω) u h φ K V K T, [u h ] = 0 E i. Th dual nod functionals that dfin th finit lmnt basis (and consquntly th dgrs of frdom; s []) for this spac ar givn by th intgral avrags (.5a) ψ (ϕ h ) = ϕ h E i, (.5b) ψ K (ϕ h ) = ϕ h w K K T, K whr th function w K (x) is th pushd-forward vrsion of th wight x 1 x on th rfrnc cll K = [ 1, 1] undr th mapping φ K : K K. 3. Furthr notation. In th sctions blow, w will discuss how to xtnd th dfinitions of th prvious sction to th cas of mshs T with hanging nods. To this nd, lt us introduc th notation w will rquir blow. Triangulation. Lt T 0 b a rgular triangulation on Ω which consists of quadrilatrals so that two closd clls K 1, K T 0 ithr shar a common vrtx, a complt dg, ar idntical, or do not touch at all. W can thn rfin it rcursivly into th targt triangulation T = T N by nsuring that T n rsults from T n 1 by rgular bisction in both dirctions of a subst of clls, and nsuring that nighboring clls nvr diffr by mor than on rfinmnt lvl (i.., ach dg has at most on hanging vrtx). 1 In th cas of Rannachr Turk lmnt [15], on chooss θ(ξ) = ξ. Howvr, for this choic, (.3) dos not hold. Instad of th paramtric vrsion of th DSSY lmnt introducd abov, thr is also a nonparamtric vrsion with four dgrs of frdom [14] such that th man valu proprty (.3) holds for gnral quadrilatrals and optimal convrgnc is attaind.

5 NONCONFORMING FINITE ELEMENTS WITH HANGING NODES jk, jl jk # jk kj K k K j jl # jk, jl + jl lj K l Fig. 1. Local notation for dgs on clls with a hanging dg. Suprscripts #, +, and to jk dnot th dgs of K j that li opposit to jk, that shar a vrtx with kj, and that do not shar a vrtx with kj but ar also not opposit to jk, rspctivly, and similarly for th suprscriptd vrsions of jl. Not th quivalncs # jk = # jl, + jk = jl, jk = + jl, jk = jl. Vrtics and dgs. For any lmnt K T, lt N (K) dnot its vrtics, and E(K) its dgs. Th sts of all nods and dgs in th triangulation T ar dnotd by N := K T N (K) and E := K T E(K), rspctivly. Furthrmor, th sts of all intrior and boundary nods and dgs ar dsignatd by N i, N b, E i, and E b, rspctivly. Hanging dgs and nods. W call placs whr clls of diffrnt rfinmnt lvls com togthr (s Figur 1) hanging dgs and th singl vrtx at thir midpoints hanging nods. In ordr to concisly dfin th finit lmnt spac V h (T) on mshs with hanging dgs, w will nd th following trms: 1. (Gnration and dscndant nighbors) Givn an lmnt K T which is obtaind by (rcursiv) rfinmnt from lmnt K 0 T 0, lt th gnration of K, dnotd by Gn(K), b th numbr of bisction stps rquird to go from K 0 to K [7]. A cll K T is calld a dscndant nighbor of K if K and K ar adjacnt and Gn(K ) > Gn(K). Th on-hanging-nod-prdg rul guarants that for any two nighbors K and K, w hav that Gn(K ) Gn(K) { 1, 0, 1}.. (Hanging nods) A vrtx z N i is calld a hanging nod if thr xists an lmnt K T such that z K\N (K). W will dnot by N H th st of all hanging nods and by N R := N i \ N H th st of all rgular (intrior) nods. 3. (Hanging dgs) An dg E i is calld a hanging dg if thr xists a hanging nod z that is not on of th nd points of. An dg E i is calld a child dg if thr xists a hanging dg such that. W will dnot by E H th st of all hanging dgs, E C th st of child dgs, and E R := E i \(E H E C ) th st of rgular dgs. 4. (Local dscription of hanging dg) Lt jk b th dg of cll K j adjacnt to a nighboring cll K k. If jk E R, thn jk is also a complt dg kj of cll K k. On th othr hand, suppos two lmnts K j, K k T ar adjacnt such that K k is a dscndant nighbor to K j ; thn jk E H and kj E C. W not that hanging nods ar concptually part of small clls along rfind dgs, whras hanging dgs blong to th larg clls. W also rmark that E i is th disjoint union of E R, E H, and E C and that N i is th disjoint union of N R and N H. W considr rgular nods N R and dgs E R to b substs of th intrior nods N i and dgs E i only sinc boundary nods and dgs will not b of any concrn in th following. W introduc furthr notation convntion as follows. Lt jk b a hanging dg. Thn lt # jk dnot th dg of K j which lis opposit to jk. Among th rmaining

6 174 W. BANGERTH, I. KIM, D. SHEEN, AND J. YIM two dgs of K j, lt + jk and jk b th dgs that do and do not shar a vrtx with kj, rspctivly (s Figur 1 for an illustration). It is worth noting that an dg of cll K j can b dnotd in svral diffrnt ways dpnding on which of its nighbors w currntly considr. Spacs and norms. Lt S b a boundd opn domain in R. Dnot L (S) and H 1 (S) th standard Sobolv spacs on S with th standard norms 0,S and 1,S. H0 1 (S) is th subspac of H 1 (S) consisting of functions whos trac on S is zro. Not that th standard sminorm 1,S is a norm on H0 1 (S) providd that th boundary of S is sufficintly smooth. (, ) S and, S dnot th L (S) and L ( S) innr products, rspctivly. If S = Ω, th subscript Ω on norms and innr products will b omittd. Avrags and jumps. Lt ω b a two- or on-dimnsional masurabl subst with ω > 0. For u L 1 (ω), dnot by ω u := 1 ω u th avrag intgral valu of u on ω ω. On an dg jk E i, w dfin th jump [u] jk of u on jk by [u] jk := u Kk u Kj. Not that th jump [u] jk dpnds not only on th location of th dg but also on its dirction. In particular, [u] kj = [u] jk. For an dg j of K j on Ω, w dfin [u] j := u Kj. W will occasionally writ [u] and omit th subscript of th dg dirction if th jump (or an intgral of it) across an dg is zro. Throughout this papr, C dnots a gnric constant for inqualitis. Th msh paramtr h is dfind by h := max K T diam(k). 4. Discrtization with th DSSY lmnt. W will us th Poisson quation (.1a) (.1b) to illustrat how w discrtiz partial diffrntial quations using th DSSY lmnt dscribd in sction and how this informs our approach for constructing hanging nod constraints for situations such as thos dscribd in sction 3. Bcaus th choic of boundary valus ar immatrial to th discussion hrin, lt us assum for simplicity that w us homognous Dirichlt boundary conditions, i.., g = 0. Thn rcall that th wak form of problm (.1) rquirs us to find u V 0 = H0 1 (Ω) so that (4.1) a(u, v) = (f, v) v V 0, whr a(u, v) = ( u, v) = u v dx. Ω Th nonconforming Galrkin approximation to (4.1) is to find u h V h,0 = V h,0 (T) so that (4.) a h (u h, v h ) = (f, v h ) v h V h,0. Hr, V h,0 (T) is th spac of all functions in V h (T) that hav man valu zro on ach boundary fac E b of th msh, and (4.3) a h (u h, v h ) = K T( u h, v h ) K. For futur us, lt us also dfin th brokn nrgy norm for functions v V h (T)+ H 1 (Ω) that is associatd with this bilinar form by v h = a h (v, v).

7 NONCONFORMING FINITE ELEMENTS WITH HANGING NODES Ky stps in a priori rror stimats on rgular mshs. Unlik for ntirly discontinuous finit lmnt spacs, th structur of V h allows us to solv this problm without th nd for additional pnalty trms to trat th discontinuity of functions in V h. Howvr, rror stimats incur a nonconformity pnalty [1]. Consquntly, if thr ar no hanging nods in th msh, th nrgy norm rror of th numrical solution consists of th usual bst-approximation rror plus th pnalty and is boundd as follows: (4.4) ( u u h h C inf ϕ h V h u ϕ h h + sup w h V h a h (u, w h ) (f, w h ) w h h For th purposs of this papr, th first of th two trms on th right-hand sid is not important as w know that it is of optimal ordr [1]. Howvr, it is ncssary to undrstand th dtails of th scond trm as it will inform us about th proprtis w will nd of th discrt spac in th prsnc of hanging nods if w want to nsur optimality of th convrgnc ordr. Intgrating its numrator by parts on ach cll and using th strong form (.1) of th quation, w s that ). a h (u, w h ) (f, w h ) = K ( u, w h ) K (f, w h ) K = K ( u f, w h ) K + n K u, w h K = E i n u, [w h ]. Hr, n K is th unit outward normal to cll K, and n dnots th unit normal vctor to dg with a dirction consistnt with that chosn for th jump [w h ]. Bcaus th jump [w h ] of a function w h V h is zro on avrag (s (.4)), w can subtract any constant from n u and lav th rsult unchangd. Spcifically, dnoting g = u, w gt a h (u, w h ) (f, w h ) E i n ( u g ), [w h ] E i u g 0, [w h ] 0,. Standard polynomial intrpolation stimats imply that (4.5a) ( ) 1/ u g 0, Ch 1/ u H(Ω), E i (4.5b) ( ) 1/ [w h ] 0, Ch 1/ w h h. E i Consquntly, a h (u, w h ) (f, w h ) Ch u H (Ω) w h h, and (4.4) provids us with an optimal convrgnc ordr in th brokn H 1 norm: u u h h Ch u H (Ω). 4.. Options for xtnsions to locally rfind mshs. Givn ths argumnts, th qustion bfor us is how w nd to xtnd dfinition (.4) of th finit lmnt spac V h (T) to th cas of mshs with hanging nods. Our xprinc with how this is don for conforming lmnts, as wll as th critical pics of th thory outlind abov, has ld us to thr possibl options that w will illustrat with th notation of Figur 1:

8 176 W. BANGERTH, I. KIM, D. SHEEN, AND J. YIM A. Rquir that (4.6) w h = w h and w h = w h, kj jk lj jk i.., that th avrag of a function w h on vry small dg kj, lj quals th avrag on th long dg jk = jl. This approach constructs constraints that only involv dgrs of frdom dfind on th parnt dg and its childrn. Howvr, it dos not provid th ky proprty that [w h] = 0 for all dgs: whil it is tru for rgular dgs E R and parnt dgs E H as a whol, it dos not hold for individual child dgs E C. Th constraint in (4.6) can b intrprtd as constraining th dgrs of frdom on th child dgs against th dgr of frdom on th hanging (parnt) dg. This is how constraints ar typically writtn for conforming lmnts and would allow th us of algorithms and data structurs that ar alrady implmntd in common finit lmnt packags. Disappointingly, howvr, this advantagous proprty is ntirly ngatd by th fact that this choic of constraint construction dos not lad to a convrgnt schm (s sction 7). B. Rquir that (4.7) [w h ] kj = 0, [w h ] lj = 0. kj lj Hr, th avrag jump [w h] = 0 is zro on all child dgs individually and consquntly also on all parnt dgs. W thrfor know that th solution satisfis th proprtis rquird in stablishing th a priori rror stimats outlind abov, and th convrgnc proof follows trivially. This formulation of constraints allows th valus on child dgs to b diffrnt and thrfor imposs a wakr constraint than th first approach abov. Unfortunatly, as w will show in th following sction, th constraints constructd this way coupl not only dgrs of frdom on a parnt dg and its childrn but in fact all dgrs of frdom on a larg cll with thos on childrn of its dgs. This can thn lad to chains of constraints that propagat byond a singl cll and consquntly rquirs significantly mor complicatd data structurs and algorithms than th usual cass ncountrd with conforming lmnts. C. Rquir that (4.8) 1 { w h + kj lj w h } = w h, jk i.., th avrag of th avrags on th child dgs quals th avrag on th larg dg. This statmnt is wakr than th on usd in (4.7) and can b intrprtd as constraining th dgr of frdom on a parnt dg against th two valus on th child dgs, in a rvrsion of th way w usually build hanging nod constraints. As w will show blow, this approach constructs constraints only coupling shap functions corrsponding to dgrs of frdom locatd on th parnt dg and its childrn. It dos not satisfy th proprty [w h] = 0 for individual child dgs E C, and consquntly w will hav to work hardr to prov convrgnc.

9 NONCONFORMING FINITE ELEMENTS WITH HANGING NODES 177 As w will s numrically in sction 7, th first of ths thr choics rsults in a suboptimal convrgnc ordr on mshs with hanging nods. W will thrfor not considr it in any mor dtail. Th scond and third turn out to b largly comparabl in accuracy, though option B sms to crat smallr L rrors in som circumstancs. On th othr hand, as alrady mntiond abov, option B has mor svr practical consquncs in implmntations and consquntly may b computationally mor xpnsiv. W will thrfor considr both options B and C in mor thortical and practical dtail in th following two sctions; w start with option B sinc its analysis is lss involvd. Rmark 4.1. Non of th thr options abov inhrntly rly on th fact that rfinmnt of clls is don by qual bisction of clls into four childrn. Rathr, th formulas ar asily adaptd to th cas whr on rfins a cll into 3 3 childrn, or whr dgs ar not bisctd at thir midpoints. This would, for xampl, facilitat th construction of gradd mshs rfind toward boundary or intrnal layrs. Rmark 4.. Th approachs abov all incorporat th tratmnt of hanging nods into th strong formulation of th discrt problm, i.., by adjusting th finit dimnsional spac. Thr ar, howvr, altrnativ approachs. For xampl, on can add pnalty trms in th styl of DG mthods for hanging dgs only (s,.g., [11]). Ths approachs ar not of intrst to th currnt papr, howvr. 5. Extnding th DSSY spac to mshs with hanging nods and dgs: Option B. Th discussions of th prvious sction motivat us to analyz in mor dtail th thr options for dfining th nonconforming spac V h (T) in th prsnc of hanging nods. W will start with th scond of th thr options (s (4.7)), as this turns out th simplst in trms of its convrgnc thory. It lads to th following spac on mshs with hanging nods and dgs: { V h (T) = u h L (Ω) u h φ K V K T, } (5.1a) [u h ] = 0 (E R E C ), { } (5.1b) V h,0 (T) = u h V h (T) u h = 0 E b. In analogy to (.5a) and (.5b), w can dfin dual functionals (5.) ψ (ϕ h ) = ϕ h (E R E H ) and ψ K as dfind in (.5b). Ths thn induc a basis {ϕ j } j=1,...,dim (Vh,0 (T)) for V h,0 (T). W can comput th dimnsions of th spacs so dfind as follows. Lmma 5.1. Th dimnsion of V h (T) is qual to E R + E H + E b + T. Th dimnsion of V h,0 (T) is qual to E R + E H + T. Proof. Sinc th spac V dfind in (.) has dimnsion fiv, th spac { Vh DG (T) = u h L (Ω) u h φ K V } K T has dimnsion 5 T. Bcaus vry rgular dg E R is part of two clls whras vry dg in E H, E C, E b is only part of on cll, and bcaus vry cll K has four

10 178 W. BANGERTH, I. KIM, D. SHEEN, AND J. YIM dgs, w know that 4 T = E R + E b + E H + E C. Consquntly, dim Vh DG (T) = E R + E b + E H + E C + T. On th othr hand, V h (T) diffrs from Vh DG (T) by th imposition of E R + E C linarly indpndnt constraints. Consquntly, dim V h (T) = dim Vh DG (T) E R E C, proving th first claim of th lmma. Th rsult for th cas of V h,0 (T) follows immdiatly from that for V h (T) by a similar argumnt, yilding dim V h,0 (T) = dim V h (T) E b. This, not coincidntally, also matchs th numbr of dual functionals w hav dfind in (5.) and (.5b). Thorm 5.. Using th spac V h,0 dfind in (5.1b), th solution of (4.) satisfis th nrgy norm stimat u u h h Ch u H (Ω). Proof. Th proof follows without chang from th on givn in [1] and outlind in sction 4. Th ky point is that th avrags of th solutions on both sids of child dgs ar qual, and consquntly th ky stps in th stimat of th nonconformity rror discussd in sction 4 continu to hold. Lt us dvot th rmaindr of this sction to a discussion of th constraints w impos in (5.1a). In th contxt of adaptivly rfind mshs with hanging nods, on typically considrs two points of viw that inform us about thortical and practical ways to dal with hanging nods: (Global basis functions) Th functionals (5.) and (.5b) dfin a basis for V h,0 (T) in such a way that thr is a basis function ϕ for ach dg (E R E H ) and a basis function ϕ K for ach cll K so that ψ (ϕ ) = δ,, (E R E H ), ψ (ϕ K ) = 0 (E R E H ), K T, ψ K (ϕ ) = 0 (E R E H ), K T, ψ K (ϕ K ) = δ K,K K, K T. If th msh T has no hanging nod, thn ach of th ϕ has support on xactly th two clls adjacnt to th dg and ϕ K has support only on K. Howvr, w will show blow that if T dos hav hanging nods, th support of basis functions ϕ associatd with an dg may in fact xtnd byond th boundaris of th clls adjacnt to. In othr words, basis functions ar nonlocal in th sns that th xtnt of thir support dpnds not only on th local gomtry of th clls adjacnt to an dg, but also byond. (Constraints) In implmntation practic, on nvr constructs th global basis functions ϕ and ϕ K. Rathr, implmntations of finit lmnt mthods on adaptivly rfind mshs with hanging nods almost always dfin intrmdiat spacs that ignor hanging nods and thn impos additional constraints during th assmbly and solution procss [16, 5, 1]. In th currnt contxt, ths intrmdiat spacs would b { Ṽ h (T) = ũ h L (Ω) ũ h φ K V K T, } (5.3a) [ũ h ] = 0 E R, { } Ṽ h,0 (T) = ũ h Ṽh(T) (5.3b) ũ h = 0 E b,

11 NONCONFORMING FINITE ELEMENTS WITH HANGING NODES 179 with dual functionals (5.4) ψ ( ϕ h ) = ϕ h E i, and ψ K as dfind in (.5b). Ths spacs and functionals ar dsignd in a way so that on can numrat dgrs of frdom on vry intrior dg, whthr it is a rgular, furthr rfind, or child dg. In particular, using this viwpoint, on dfins a basis function both for ach hanging dg as wll as its two childrn. A function ũ h = j Ũj ϕ j using this basis is thn obviously part of Ṽh,0(T), but not ncssarily V h,0 (T). To forc it into th lattr spac, on nds to impos constraints [ũ h ] = 0 E C. This condition can b rwrittn in trms of th xpansion of cofficints Ũj: (5.5) C j Ũ j = 0 E C, j whr (C j ) = ( [ ϕ ) j] is a spars matrix of siz EC dim(ṽh,0). With such constraints in hand, on thn builds a (rank dficint) linar systm ÃŨ = F using th basis for Ṽh,0 and solvs it subjct to constraint (5.5) to obtain a solution u h = j Ũj ϕ j V h,0. Whil it is usful to think in trms of th global basis functions ϕ j whn analyzing a discrtization on a msh with hanging nods, th point of viw of constraints prvails whn considring implmnting a particular lmnt on such a msh. This is so bcaus both th assmbly of linar systms as wll as th computation of th constraint matrix (C j ) is most asily implmntd using th basis functions ϕ, ϕ K : ths ar th only nonzro basis functions on a cll K, rgardlss of whthr any of th dgs of K is in E C. Furthrmor, knowing (C j ) allows us to construct th basis {ϕ j } from th basis { ϕ j }; s [1]. For ths rasons, w will in th following discuss th proprtis of (C j ) in mor dtail. Obviously, (5.5) dfins on constraint pr child of ach hanging dg. As an xampl, lt us considr dg kj btwn larg cll K j and child nighbor K k from Figur 1. Thn th constraint that corrsponds to this dg rads ) (Ũkj ϕ kj = kj kj ) (Ũjk ϕ + Ũ # jk ϕ jk # + Ũ + ϕ jk jk + + Ũ ϕ jk jk + ŨK j ϕ Kj. jk By th dfinition of ϕ kj, w hav that kj ϕ kj = 1, and w can rwrit th constraint in th usual form that allows us to liminat a dgr of frdom dfind on a child dg in trms of othr dgrs of frdom using to th following rlation: [ ] [ ] Ũ kj = ϕ jk Ũ jk + ϕ # Ũ jk # jk kj kj }{{}}{{} =1 =0 [ ] [ ] [ ] + ϕ + Ũ jk + + ϕ jk Ũ jk + ϕ Kj Ũ Kj jk kj kj kj }{{}}{{}}{{} = 1 4 = 1 =0 4 = Ũ jk + 1 4Ũ + 1 jk 4Ũ jk.

12 1730 W. BANGERTH, I. KIM, D. SHEEN, AND J. YIM Fig.. Dual points of viw for chains of constraints. Lft: Th cofficint Ũ corrsponding to a dgr of frdom dfind on an dg E C may b constraind against a tlscoping st of othr dgrs of frdom. Th figur shows th supporting dgs and clls of th complt st of dgrs of frdom against which Ũ is dfind. Right: Th xtndd support of th global basis function ϕ for on E R. Not that th factors in th first, third, and fourth squar brackts on th right ar not zro for th DSSY lmnt bcaus, for xampl, th shap function ϕ + has man jk valu zro on th long dg jk but not on ach of th childrn kj, lj of jk. This is diffrnt from th usual conforming spacs on mshs with hanging nods whr ach dgr of frdom dfind on a child of a hanging dg E C is constraind only in trms of dgrs of frdom dfind on th parnt of, but not othr parts of th adjacnt (larg) cll. This obsrvation is important bcaus, of cours, ithr of th dgrs of frdom corrsponding to th cofficints Ũ +, Ũ may itslf li on th child of a hanging jk jk dg and consquntly b constraind itslf. In othr words, in actual computations, w may obtain chains of constraints. In ordr to liminat th constraind dgr of frdom corrsponding to th cofficint Ũ kj w will thrfor hav to rcursivly xpand all dgrs of frdom that appar on th right-hand sid in trms of th dgrs of frdom thy may b constraind against. This is similar to situations on may find in hp adaptiv mthods in thr dimnsions (s [1]) and rquirs th us of mor complx data structurs and algorithms than on might us for mshs with hanging nods and usual finit lmnts. 3 Rmark 5.3. Th lngth of ths chains of constraints is boundd by th diffrnc btwn th maximal and minimal gnration of clls in T bcaus it always constrains dgrs of frdom on mor rfind clls against dgrs of frdom on lss rfind nighboring clls. Similarly, th rcursiv xpansion of constraints nvr lads to circular constraints as long as T originats from a rgular msh T 0 without hanging nods as discussd in sction 3. To illustrat th xtnt to which chains of constraints can xpand th support of basis functions, Figur shows both all othr dgrs of frdom on particular dgr of frdom dfind on a child dg is constraind against (lft), as wll as th support of an unconstraind basis function in th basis V h (right). It is worth pointing 3 Th fact that th constraint wight [ ϕ # kj ] for th shap function dfind on th opposit jk dg of cll K j happns to b zro is a consqunc of putting th hanging nod at th midpoint of dg jk. Had w chosn an unqual subdivision of th dg, th xpansion cofficint Ũ would kj also b constraind against Ũ # with a nonzro wight. jk

13 NONCONFORMING FINITE ELEMENTS WITH HANGING NODES 1731 out that to gt chains of constraints, it is ncssary to hav vrtics in th msh with adjacnt clls that diffr in gnration by at last two. On th othr hand, widly usd msh smoothing tchniqus spcifically prohibit this, by flagging additional clls for rfinmnt; in ths cass, chains of constraints cannot happn as long as rfinmnt biscts dgs at th midpoint. 6. Extnding th DSSY spac to mshs with hanging nods and dgs: Option C. Th discussions of th prvious sction showd that option B for imposing hanging nod constraints (s sction 4.) allowd for a trivial xtnsion of th proof of convrgnc compard to th original on for rgular mshs. On th othr hand, it lads to constraints that ar vry diffrnt in structur compard to thos on typically ncountrs in conforming finit lmnt mthods. This obsrvation motivats us to considr th third option suggstd in sction 4., namly, to rquir that 1 { w h + kj lj w h } = w h jk for ach hanging dg jk. In th following subsctions, w will thortically invstigat th convrgnc proprtis of this approach and thn considr th structur of ths constraints in analogy to th discussion in th prvious sction Convrgnc thory. Th formulation of constraints at hanging dgs considrd in this sction implis that [w h] = 0 for all parnt dgs E H, but not th ky proprty [w h] = 0 for child dgs E C ; this proprty mad th convrgnc proof in th prvious sction trivial. Nvrthlss, w can still show th following rsult. Thorm 6.1. Lt { V h (T) = u h L (Ω) u h φ K V K T, { } V h,0 (T) = u h V h (T) u h = 0 E b. Thn th solution of (4.) satisfis th nrgy norm stimat u u h h Ch u H (Ω). } [u h ] = 0 (E R E H ), As in th cas of option B in sction 5, it is only ncssary to considr th numrator of th consistncy rror trm. It can hr b split into th following trms: a h (u, w h ) (f, w h ) = n u, [w h ] + I. E R E H Th trms on rgular dgs ar tratd as in th cas without hanging nods, for which w rfr to sction 4 and [1]. Consquntly, w only hav to stimat th trms on hanging dgs, I jk = n jk u, [w h ] jk jk = n jk u, w h n jk u, w h n jk jk u, w h. kj lj As bfor, n is th normal to dg with dirction consistnt with that chosn to dfin th jump [w h ].

14 173 W. BANGERTH, I. KIM, D. SHEEN, AND J. YIM To show th rsult of th thorm, lt us dfin thr avrag valus: { g jk = u, w jk = w h and w kj,lj = 1 } w h + w h. jk jk kj lj Th choic of constraints implis that w jk = w kj,lj, or quivalntly jk [w h ] jk = 0, for ach hanging dg jk. Consquntly, I jk = n jk ( u g jk ), [w h ] jk jk = n jk ( u g jk ), ( ) w h w jk n jk ( u g jk ), ( ) w h w kj,lj jk kj n jk ( u g jk ), ( ) w h w kj,lj lj u g jk w h w 0,jk jk 0,jk + ιj u g jk 0, ιj 1/ ιj 1 wh w kj,lj Th difficulty to ovrcom hr is that in th last trm, w h w kj,lj dos not hav man valu zro ovr th child dg ιj, ι {k, l} w intgrat ovr and consquntly dos not provid us with th ncssary powr of th msh siz. For th following, lt us dnot th diamtr of th lmnt K T and of th dg E by h K and h, rspctivly. Thn th following thr auxiliary stimats hold from ssntial intrpolation stimats (th first on mirroring (4.5a)): u g jk Ch 0,jk 1/ jk u,kj, wh w 0,jk jk Ch 1/ jk w h 1,Kj, 1/ u h ιj gjk C 1/ h ιj u,k ι. 0, ιj Th main difficulty of th rror analysis is to show th following lmma. Lmma 6.. (6.1) h 1 ιj wh w kj,lj 0, ιj 1/ C w h 1,K ι For th proof of this lmma, lt us first introduc som notation and auxiliary rsults. To this nd, lt K k := [0, 1] [0, 1], Kl := [0, 1] [ 1, 0] b th two rfrnc squars and F ι : Kι K ι b th bilinar transformation onto K ι for ι {k, l}. S Figur 3 for illustration. Thn th picwis bilinar transformation F is dfind by 1/. 0, ιj 1/.

15 NONCONFORMING FINITE ELEMENTS WITH HANGING NODES 1733 ^ kj K ^ k F k kj K k ^ Γ σ^ Γ σ ^lj K ^ l lj K l F l Fig. 3. Picwis bilinar mappings F ι : Kι K ι, ι {k, l}, btwn th rfrnc squars and gnuin quadrilatrals. x = F( x) := { F k ( x), x K k, F l ( x), x K l. It is usful to dfin th combind rfrnc domain K kl := K k K l and th associatd domain K kl := K k K l. Dnot by ê ιj th dg of K ι associatd with ιj for ι {k, l} and by σ th dg common to K k and K l. W will dsignat by Γ = {0} [ 1, 1] th part of th boundary of K kl. σ and Γ will man th imags undr F of σ and Γ, rspctivly. Gomtrically, Γ is idntical to jk and jl in Figur 1. W invok th following standard rsults drivd using shap rgularity of T. Lmma 6.3. Thr xist constants c, C > 0 such that, for vry lmnt K T, ch K dt x (6.a) x Ch K, x m (6.b) x n Ch K, m, n = 1,. Lmma 6.4. For givn w h V h,0 (T), dfin ŵ h := w h Kkl F L ( K kl ). Thn, th following statmnts hold: 1. [ŵ h ] σ = 0,. ŵ h = w kj,lj. σ W ar now rady to prov Lmma 6.. Trac and Poincaré Fridrichs inqualitis for picwis H 1 functions on th rfrnc domains [] imply that w can stimat as follows: h 1 ιj wh w kj,lj 0, ιj C C[ ŵh w kj,lj = C Γ [ ŵ h w kj,lj 0,ê kj + ŵ h w kj,lj 0,ê lj ] 0, K + ŵh w kj,lj k 1, K + ŵh w kj,lj k ŵh w kj,lj 0, K kl + C = C ŵh w kj,lj 1, K ι + ŵ h 1, K ι C ( ŵ h w kj,lj σ w h 1,K ι. 1, K ι 0, K l + ŵh w kj,lj ) + ( Γ [ŵ h w kj,lj ] σ dŝ ŵ h w kj,lj 1, K l ] ) dŝ

16 1734 W. BANGERTH, I. KIM, D. SHEEN, AND J. YIM This provs Lmma 6.. It is worth noting that th inquality constant in th proof dpnds on th shap rgularity of T only. All of this togthr now also allows us to finish th proof of Thorm 6.1. With th abov lmmas, w obtain I jk C h jk u,kj w h 1,Kj + 1/ 1/ h ιj u,k ι w h 1,K ι C K ω( jk ) h K u,k 1/ K ω( jk ) w h 1,K 1/, whr ω( jk ) = {K j, K k, K l } is th st of lmnts containing all or parts of jk as an dg. This yilds th following, ordr-optimal bound for th consistncy rror: ( 1/ a h (u, w h ) (f, w h ) sup C h K u w h V h w h,k). h K T 6.. Structur of th constraints. Th constraints discussd in this sction only coupl dgrs of frdom on th hanging, parnt dg and its childrn, but not thos dfind on adjacnt dgs or th intrior of clls. This is diffrnt from th constraints of th prvious sction; in particular, w no longr hav to dal with chains of constraints. Using th approach of th currnt sction, w hav only on constraint pr hanging dg that contains th dgrs of frdom for both child dgs. Concptually, this can b intrprtd as constraining th dgr of frdom on th parnt dg against thos on all childrn, a rvrsal of th situation for conforming lmnts whr w constrain th dgrs of frdom on ach child dg sparatly against thos on th parnt dg. Othr than th fact that this dos not follow th usual pattrn, implmnting ths constraints in softwar dos not prsnt any undu burdns A comparison with th othr options. Proving optimal convrgnc ordr for option C rquird showing that th st of functions satisfying th constraints a (4.8) yilds optimal ordr for th nonconformity rror, sup h (u,w h ) (f,w h ) wh V h w h, in h (4.4). On th othr hand, comparing (4.8) with (4.6) and (4.7) shows that functions that satisfy constraint options A and B also satisfy th constraints of option C. In othr words, th function spacs dfind by options A and B ar smallr than thos of option C. It is thn obvious that th nonconformity rror must also b of optimal ordr for options A and B. On th othr hand, w will show xprimntally in th nxt sction that option A dos not convrg. This is bcaus for this option, th bst-approximation trm in (4.4) is not of optimal ordr. This is asy to s: globally linar functions ar not in th discrt spac dfind by constraint (4.6) if hanging nods ar prsnt. 7. Numrical rsults. In this sction, w will numrically valuat th prformanc of th thr options to impos constraints for th DSSY lmnt outlind in sction 4.. As provn in sctions 5 and 6, th scond and third options yild convrgnt schms of optimal ordr, but it is unclar which on may b bttr in absolut trms. W hr invstigat this qustion numrically and also show that th first option to dfin constraints dos not yild a schm that convrgs in th nrgy norm.

17 NONCONFORMING FINITE ELEMENTS WITH HANGING NODES 1735 Fig. 4. A msh with th maximal numbr of hanging nods. Hr, vry intrior dg contains a hanging nod. Constraint construction option A Constraint construction option B Constraint construction option C Constraint construction option A Constraint construction option B Constraint construction option C Enrgy norm rror 0.1 L_ rror # of dgrs of frdom # of dgrs of frdom Fig. 5. Maximally bad mshs, smooth solution: Convrgnc of th solution u h toward th xact solution u on a squnc of mshs with th maximal numbr of hanging nods, using th thr diffrnt options to construct constraints discussd in sction 4.. Lft: Th nrgy norm rror u u h h. Both options B and C yild a convrgnc ordr of O(N 1/ ) = O(h). Right: Th L norm rror u u h 0, convrging as O(N 1 ) = O(h ) Maximally bad mshs, smooth solution. Th worst msh as far as hanging nods ar concrnd is on in which vry othr cll is rfind, starting from a uniformly rfind msh; s Figur 4. W first invstigat th convrgnc of th numrical solution using a tst cas with th smooth xact solution u = cos(πx) cos(πy) on th L-shapd domain Ω = ( 1, 1 ) \[0, 1 ]. Figur 5 shows both th nrgy and L norm rrors for th thr options of imposing hanging nod constraints. Options B and C both show optimal ordr convrgnc in both norms, and th rrors ar in fact not substantially diffrnt. Option A dos not convrg in ithr norm. Th fact that it dos not convrg in th nrgy norm may not b surprising: Figur 6 shows th solution (for th nonsmooth tst cas discussd in th nxt sction) and dmonstrats that rquiring th valus along both child dgs to b th sam yilds a tiltd numrical solution whnvr th drivativ of th xact solution in th dirction of th dg is nonzro. Consquntly, th gradint of th numrical solution dos not convrg to that of th xact solution. That th L norm dos not convrg at any apprciabl rat (it dos dcras vry slowly) is mor surprising. A closr invstigation shows that th numrical solution is consistntly smallr than th xact on, suggsting that th imposition of ths constraints lads to a discrt problm that is too rigid. For th msh usd in th currnt sction, constraints ar not chaind bcaus no cll has both hanging dgs and dgs that ar childrn of hanging dgs of othr

18 1736 W. BANGERTH, I. KIM, D. SHEEN, AND J. YIM Fig. 6. Maximally bad mshs, singular solution: Solutions on th sam msh using options A (lft) and C (right) for th imposition of constraints. Th solution for option B looks ssntially th sam as th pictur on th right, with only marginal visibl diffrncs for th clls at th rntrant cornr. Color and lvation both show th valu of u h (x) at a point x Ω. clls. Nonthlss, th thr diffrnt options of daling with hanging nods yild vry diffrnt constraints that, if rsolvd, yild matrics with diffrnt numbrs of nonzro ntris. To illustrat this, considr a msh similar to th on in Figur 4 but rfind four mor tims globally bfor rfining vry othr cll. Th associatd finit lmnt spac has 393,16 dgrs of frdom bfor liminating constraind ons. Th thr options of constructing constraints thn yild matrics with,651,130, 4,996,106, and 3,36,864 ntris in th sparsity pattrn. 4 Whil w know that option A is not viabl, option B rsults in a matrix with about 50% mor ntris than option C. Consquntly, solvrs will typically also rquir mor CPU tim using this approach. 7.. Maximally bad mshs, singular solution. A mor intrsting tst cas is to solv a problm with a singular solution on such maximally bad mshs. W again us th L-shapd domain introducd abov and choos right-hand-sid and boundary valus in such a way that w rproduc an xact solution that in polar coordinats rads u(r, θ) = r 3 sin ( 3 ( θ π ) ), π θ π. Th numrical solutions for options A and C for th imposition of constraints ar shown in Figur 6. Th fact that for option A all small clls ar tiltd to accommodat th constraint that th avrag valus on both child dgs ar qual xplains why w do not s convrgnc in th nrgy norm. (A similar pictur, though far lss obvious, could also hav bn mad for th prvious xampl.) Givn th singularity at th origin, w cannot xpct to obtain optimal convrgnc ordrs. Indd, as shown in Figur 7, th convrgnc ordr is rducd. Options B and C do not diffr in noticabl ways in nrgy norm. Thr is a small diffrnc in th L norm whos origin w hav not bn abl to lucidat; in particular, th convrgnc ordr for option B dpnds on whthr th chckrboard msh is cratd by rfining th whit filds (O(h 1.6 )) or th black filds (O(h 1.5 )), whras option C dos not show any diffrnc in this rgard Fully adaptiv mshs, singular solution. W conclud our numrical xprimnts by comparing rsults obtaind for th singular solution tst cas abov, but using mshs that ar gnratd by an a postriori rror stimator that taks 4 W hav vrifid that asymptotically, th numbr of ntris in th sparsity pattrn and th numbr of nonzro ntris in th matrix ar th sam.

19 NONCONFORMING FINITE ELEMENTS WITH HANGING NODES Constraint construction option A Constraint construction option B Constraint construction option C 0.01 Constraint construction option A Constraint construction option B Constraint construction option C Enrgy norm rror 0.1 L_ rror # of dgrs of frdom # of dgrs of frdom Fig. 7. Maximally bad mshs, singular solution: Convrgnc of th solution u h toward th xact solution u on a squnc of mshs with th maximal numbr of hanging nods, using th thr diffrnt options to construct constraints discussd in sction 4.. Lft: Th nrgy norm rror u u h h. Both options B and C yild a convrgnc ordr of O(N 1/3 ) = O(h /3 ). Right: Th L norm of th rror. Hr, option B yilds a convrgnc ordr of O(N 0.81 ) = O(h 1.6 ) and option C a convrgnc ordr of O(N 0.69 ) = O(h 1.38 ). Option A, which dos not convrg in th nrgy norm, still yilds a convrgnc ordr of O(N 0.5 ) = O(h 1 ) in th L norm. 1 Constraint construction option A Constraint construction option B Constraint construction option C 0.01 Constraint construction option A Constraint construction option B Constraint construction option C Enrgy norm rror 0.01 L_ rror # of dgrs of frdom # of dgrs of frdom Fig. 8. Fully adaptiv mshs mshs, singular solution: Convrgnc of th solution u h toward th xact solution u on a squnc of adaptivly rfind mshs, using th thr diffrnt options to construct constraints discussd in sction 4.. Lft: Th nrgy norm rror u u h h. Option B yilds a convrgnc ordr of O(N 0.51 ) and option C a convrgnc ordr of approximatly O(N 0.5 ). Right: Th L norm of th rror. Both options B and C yild a convrgnc ordr of approximatly O(N 1 ). into account both th rsiduals of th discrt solution as wll as th nonconformity [8, 7, 10]. Bcaus th first option for nforcing constraints yilds a solution that violats th assumptions undrlying th rror stimator, w obtain nonsnsical mshs, and thrfor omit this approach from th currnt sction. Th gnratd mshs obtaind from options B and C show th xpctd rfinmnt toward th rntrant cornr whr th solution is singular and hav th familiar apparanc for such mshs. Figur 8 shows th convrgnc history for all approachs. Both options B and C yild ssntially th sam rrors. Finally, w show in Figur 9 th growth in th numbr of nonzro ntris in th systm matrix aftr application of constraints. This is rlvant bcaus, unlik th xampls on th mshs of th prvious subsctions, th currnt msh consists of clls on many lvls; consquntly, option B may gnrat chains of constraints that hav th potntial to crat many mor nonzros in th matrix. In fact, on can constru squncs of mshs in which th numbr of nonzros grows fastr than O(N) whr N is th numbr of dgrs of frdom. On th othr hand, th figur shows that in practic, th numbr of nonzros avrags around 7 pr

20 1738 W. BANGERTH, I. KIM, D. SHEEN, AND J. YIM 1+07 Constraint construction option B Constraint construction option C Numbr of nonzro matrix ntris # of dgrs of frdom Fig. 9. Fully adaptiv mshs mshs, singular solution: Numbr of nonzro ntris in th matrix as a function of th numbr of unknowns in th problm. row for option C, and slightly gratr than 7 for option B. In othr words, thr is no noticabl diffrnc in computational cost btwn th two approachs. This can b xplaind by th fact that on typical, adaptivly rfind mshs, fw hanging dgs giv ris to long constraint chains such as thos shown in Figur. In fact, in our computations, w had no cass of chaind constraints at all, sinc no vrtics xistd with nighboring clls whos rfinmnt lvls diffrd by two or mor. 8. Conclusions. For nonconforming finit lmnt mthods, th construction of hanging nod constraints is not as obvious as for conforming mthods. In this papr, w hav prsntd a numbr of options for how hanging nods can b tratd for on nonconforming lmnt, namly, th DSSY lmnt. W idntifid two diffrnt stratgis both rathr diffrnt from th structur of constraints typically ncountrd for conforming mthods for which w could prov thortically that thy convrg, and also dmonstrat this using numrical xprimnts. Thy diffr in thir implmntation complxity, but not substantially in th accuracy thy yild. W xpct that considrations similar to thos discussd hrin can also guid th drivation of constraints for othr nonconforming lmnts, as wll as for hanging nods gnratd by indpndnt rfinmnts [3]. REFERENCES [1] W. Bangrth and O. Kaysr-Hrold, Data structurs and rquirmnts for hp finit lmnt softwar, ACM Trans. Math. Softwar, 36 (009), pp. 4/1 4/31. [] S. Brnnr and R. Scott, Th Mathmatical Thory of Finit Elmnt Mthods, Txts in Appl. Math., 15, Springr, Nw York, 007. [3] S. Brnnr and L.-Y. Sung, Picwis H 1 functions and vctor filds associatd with mshs gnratd by indpndnt rfinmnts, Math. Comp., 84 (015), pp [4] Z. Cai, J. Douglas, Jr., J. E. Santos, D. Shn, and X. Y, Nonconforming quadrilatral finit lmnts: A corrction, Calcolo, 37 (000), pp [5] G. F. Cary, Computational Grids: Gnration, Adaptation and Solution Stratgis, Taylor & Francis, Nw York, [6] C. Carstnsn and J. Hu, A unifying thory of a postriori rror control for nonconforming finit lmnt mthods, Numr. Math., 107 (007), pp [7] C. Carstnsn and J. Hu, Hanging nods in th unifying thory of a postriori finit lmnt rror control, J. Comput. Math, 7 (009), pp [8] C. Carstnsn, J. Hu, and A. Orlando, Framwork for th a postriori rror analysis of nonconforming finit lmnts, SIAM J. Numr. Anal., 45 (007), pp [9] M. Crouzix and P.-A. Raviart, Conforming and nonconforming finit lmnt mthods for solving th stationary Stoks quations, RAIRO Math. Modl. Numr. Anal., R-3 (1973), pp