A posteriori error estimation for standard finite element analysis

Al hough hese es ima o s ha e been ound o be obus , see [2] and [5], he supe con e gence p ope ies a e only p o ed in e y ew cases [6]. Ac ually, he choice o he sample poin s used in he eco e y echnique does no al e he pe o mance o he es ima o [2]. Supe con e gence is no obse ed in meshes mixing di e en elemen ypes, he e o e lux p ojec ion es ima es should no be used wi h such meshes. Mo eo e , he eco e ed solu ion usually shows a poo quali y nea he bounda ies. Ne e heless, lux p ojec ion es ima o s a e he mos popula and hey ha e been implemen ed in many comme cial FE codes because o hei easy implemen a ion and p ac ical e iciency. Flux p ojec ion es ima es can be compu ed using only he exis ing amewo k o any classical FE code. Residual ype es ima o s we e i s in oduced by Babuska and Rheinhold in [7]. The esidual o he FE solu ion is used as a sou ce e m in local bounda y alue p oblems associa ed wi h he e o . These es ima o s a e de ined by wo ac o s: he selec ion o he local in e pola ion space (usually a bubble space o a p- e ined e e ence solu ion) and he app oxima ion o he bounda y condi ions by spli ing he lux jump ac oss he edges o he elemen s. Indeed, lux jump can be hough as a weak esidual. These kind o es ima o s ha e a solid heo e ical backg ound bu hei esul s ha e been ound o be less obus han lux-p ojec ion es ima es [2]. The bigges pa o he compu a ional cos o hese es ima o s is aken by he lux jump compu a ion and he lux spli ing p ocedu e. The implemen a ion o hese es ima o s is no widesp ead. This is p obably because hey need compu a ion o speci ic alues which a e useless o he gene al FE analysis. Some es ima o s a e di icul o classi y in o one o hese g oups. The es ima o s in oduced by Lade eze in [8] and Oh subo and Ki amu a in [9] build a egula ized lux ield and he e o e hey can be pu in o he i s class bu hey sol e local p oblems associa ed wi h he esidual and, consequen ly, hey can also be seen as esidual ype es ima o s (see [2]). These es ima o s p esen he usual disad an ages o he second amily. The es ima o s de ined by E ikkson and Johnson in [10] canno be classi ied in o one o he abo e g oups. They a e known as ex apola ion ype es ima o s. These es ima o s a e based on he app oxima ion o cons an s appea ing in he a p io i e o bounds. Bu hey a e a ely implemen ed in comme cial codes. This is p obably due o he same easons applying o esidual ype es ima o s. A new app oach o esidual ype es ima o s is in oduced he e. The es ima es a e seen as p ojec ions on subspaces o locally-suppo ed unc ions, hese p ojec ions a e compu ed by sol ing local p oblems. The compu a ion o lux jumps ac oss he edges is p ecluded, he e o e he main disad an ages o esidual ype es ima o s do no hold. Mo eo e , he es ima o can be applied in any kind o meshes, e en i di e en elemen ypes a e mixed. The ob ained es ima e is a lowe bound o he ac ual e o . In addi ion, since inpu s o local p oblems a e he same as hose o he s anda d p oblem, he de ini ion and he compu a ion o he es ima o a e easily in eg a ed in a ini e elemen code. The emainde o he pape is o ganized as ollows: Sec ion 2 p esen s he model p oblem and no a ion. Sec ion 3 is de o ed o e o p ojec ion compu a ion and some local p ope ies o p ojec ions. In Sec ion 4 an in e io elemen wise es ima o is de ined. Sec ion 5 deals wi h he p oblem o imp o ing he es ima ion by de ining new p ojec ion spaces in a con enien way. In Sec ion 6 we show some examples and demons a e he e ec i eness o he es ima o . 2. Model p oblem Le us conside an ellip ic p oblem o e a bounded domain n C [ij2 wi h a piecewise smoo h bounda y I' di ided in o wo pa s I',, and �- The bounda y alue p oblem is de ined in s ong o m as and - '· (A 'u) +bu = in il, u =O on� (A 'u) · n = gn on I',, , (la) (lb) (le) whe e u is de ined on n and needs o be app oxima ed. The Di ichle bounda y condi ion ( 1 b) is aken homogeneous, bu , due o he linea i y o ( 1 a) his is no a loss o gene ali y, he examples will p esen non-homogeneous Di ichle bounda y condi ions. In o de o simpli y he p esen a ion, a scala case is 2 conside ed in he heo e ical de elopmen o he es ima o . Howe e , he ex ension o ec o ial p oblems is s aigh o wa d and plane mechanical examples wi h wo deg ees o eedom a e shown in Sec ion 6. In o de o s udy he weak o m o he p oblem, equi alen o he s ong o m ( 1 ), se e al de ini ions a e needed. Le H 1 ( l) deno e he usual Sobole space o unc ions wi h squa e-in eg able de i a i es and le H � ( l) be such ha H � ( l) = { w EH 1 ( l) J w = 0 on �}. The solu ion u o ( 1) is also he weak solu ion o he ollowing in eg al p obl�m: ind u in H � ( l) such ha d a(u, ) = l( ) o all in H � ( l), whe e a(·, ·) and /(·) a e gi en by d and a(u, ) := J (Vu· AVu + bu ) dil {) l( ):= J d l + J gn dI'. {) 1;, (2) (3) (4) The Gale kin FE solu ion uh belongs o a ini e dimensional subspace 'Vh o H � ( l) and e i ies (2) o all d in 'Vh. The goal o e o es ima ion is o app oxima e a measu e o he e o e, de ined as e := u - uh. I A is a symme ic posi i e de ini e ma ix and b is non-nega i e, see Eq. ( la), hen he bilinea o m a(·, ·) is also symme ic posi i e de ini e and, he e o e, i is a scala p oduc . The no m 11·11 induced by a(·, ·) is called he ene gy no m; o any , ll l := a( , )112 . I Eq. (1) models a physical p oblem, Jl ll ep esen s he ene gy associa ed o he s a e o he sys em desc ibed by . The ene gy no m will be used o measu e he e o e. The FE solu ion uh is he p ojec ion o he exac solu ion u on 'Vh acco ding o he scala p oduc a(·, · ), in ha sense Gale kin me hod is said o be op imal. In o de o ha e in o ma ion abou he dis ibu ion o e o o e l, local es ic ions o 11·11 will be used. Fo a gi en subdomain nk o n he associa ed local no m o he e o e is llellk: = [a/e, e)] 112: [ L, (Ve · AVe + be2) d l J 112 (5) In he ollowing sec ions, subdomains lk a e chosen as he elemen s o he mesh which de ines he app oxima ing space °/l'h (compu a ional mesh). 3. E o p ojec ions An e o es ima e should p o ide local in o ma ion o he e o in o de o desc ibe i s spa ial dis ibu ion. The compu a ion o an e o es ima e mus also be local; ha is he compu a ion o a local es ic ion o he e o in a subdomain lk only uses in o ma ion in i and, e en ually, in i s neighbo hood. O he wise, e o es ima ion becomes oo expensi e and i could be cheape o use a e ined mesh in o de o ob ain a mo e accu a e solu ion. The local es ima o p esen ed in he ollowing sec ion is made independen o he neighbo elemen s by o cing he app oxima ion o e o anish on he bounda y. Then, in Sec ion 5 he es ima o is imp o ed, p ecluding he imposi ion o his a i icial condi ion. be o e en e ing in he de ailed de ini ion o a local es ima o , we discuss e o p ojec ions which a e he a ionale o he p oposed es ima o . Since he weak equa ion (2) is well posed and he o m a(·, ·) is assumed o be linea on he i s a gumen , he e o e is he only elemen o H� ( l) e i ying d a(e, ) = i( ) := l( )-a(uh, ) (6) o all in H� ( l). Eq. (6) is he weak global equa ion o he e o . Thus, al hough he e o is no known, i s d scala p oduc wi h any gi en unc ion may be compu ed. Tha is, he p ojec ion o e on any subspace o H � ( l) can be e alua ed. d 3 Le Vic be a subspace o H � ( l) o unc ions ha ing hei suppo in l k: hey anish elsewhe e and, consequen ly, on he bounda y � {)k (Vic is included in H�( lk)). The subspace Vk is gene a ed by a basis o in e pola ion unc ions £!lJ = {N1, ••• , Nn}. Re e ence o index k in he no a ion o he elemen s o £!lJ is omi ed when i is no necessa y. The p ojec ion o e on Vic is deno ed by Bk, he elemen Bk o Vic is ep esen ed by he column ec o [ B] g/j o i s componen s on £!1J, he scala p oduc a(·, ·) is ep esen ed by ma ix K wi h gene ic � m Kij = a(N;, N) (i, j = 1, ... , n) and he linea o m !(·) is ep esen ed by ec o [[] �i h gene ic e m [l]j l(�). Finally, he p ojec ion Bk is ob ained by sol ing he linea sys em o equa ions K[B]g/j [[]. (7) Then, he ene gy no m o Bk can be compu ed in a s aigh o wa d manne by he exp ession (8) The se o subspaces Vic is o hogonal wi h espec o a(·,·) because all subdomains nk ( he ni e elemen s) ha e in e sec ion wi h measu e ze o. Bessel's inequali y applies and he no ms o p ojec ions Bk e i y m m L llekll2 = L a(Bk, Bk)� a(e, e) = llell2 , k=l k=l whe e m s ands o he numbe o elemen s in he mesh. (9) The bound gi en by (9) holds e en locally. Indeed, as Bk is he p ojec ion o e on Vk acco ding o he global o m a(·, · ), he di e ence e Bk is o hogonal o Bk, i.e. a(e - Bk, Bk)= 0. Mo eo e , his o hogonali y condi ion also s ands o he local o m ak(·, · ), i.e. aie Bk, Bk)= 0, because he suppo o Bk is included in ilk. Thus, ak(e, e) ak([e -Bk]+ Bk, [e -Bk]+ Bk) = aiBk, Bk)+ ak(e Bk, e Bk) and a lowe bound o he local measu e o he e o is eco e ed (10) Acco dingly, we deduce ha a amily o subspaces Vic o H�}il), de ined o e disjoin subdomains nk (o wi h null in e sec ion), allows us o compu e local p ojec ions Bk such ha : • he no m o each Bk is a lowe bound o he local no m o he ue e o , see Eq. ( 10), and can be aken as a local e o es ima o , •de ning B as he sum o he Bk, m B·=" B . L., k k=l (11) he global no m o B is easily compu ed om he local p ojec ions and ge s a lowe bound o he global measu e o he e o llell m liell2 = L liekll2 � llell2 , (12) k=l •bo h global and local app oxima ions, B and he Bk's, would be mo e accu a e as he space V, gene a ed by he V/s, V := V1 EBV2 EB··· E9Vm, is close o H�}il) (asymp o ic beha io o Bessel inequali y (9) is gi en by Pa se al's equali y). The global app oxima ion B is he p ojec ion o e on V. The unc ions o each subspace Vic anish on he bounda y o ilk. Then, he global app oxima ion B associa ed wi h a amily o Vic's, de ined om a pa i ion o n in o subdomains {lk, akes ze o alues a he poin s lying on he bounda y o he il/s. Those a e called hidden poin s o he global app oxima ion. In ac , we a e a i icially o cing B o be ze o a hese poin s. The space V will ne e app oxima e enough H� ( l) i many poin s o he d domain a e hidden. In he ollowing sec ions we de ne new amilies o subspaces ela ed o o he pa i ions o n in o de o educe hese hidden poin s. 4 4. De ini ion o a local es ima o . In e io esiduals The de ini ion o he local subspaces Vi, cha ac e izes he amily o unc ions ek. The e/s a e aken as app oxima ions o he local e o . Tha is, app oxima ions o he e o inside each elemen {lk whe e he subspace Vi, is de ined. Local subspaces o esidual ype es ima o s ha e been ypically de ined ollowing a p- e inemen [11-13) o an h- e nemen s a egy [14). The p esen es ima o can ollow bo h app oaches. Fo he sake o simplici y we only show an h- e inemen app oach. The in e pola ion unc ions Ni gene a ing Vi, a e de ined by a disc e iza ion o he elemen lk. We call hese meshes elemen a y submeshes. In o de o make compu a ions sys ema ic and o ha e a simple implemen a ion, we desc ibe an elemen a y submesh o e a e e ence elemen . Gene ic elemen a y submeshes a e buil applying he isopa ame ic ans o ma ion o his e e ence elemen a y submesh, see Fig. 1. Then, he in e pola ion unc ions Ni a e associa ed wi h he nodes o he elemen a y submesh disc e izing ilk. The nodes o he elemen a y submesh lying on he bounda y o {lk a e no in oduced in he de ini ion o Vi, because he unc ions o Vi, mus anish on he bounda y. The assembly o all he elemen a y submesh builds up a e ined mesh disc e izing he whole domain l. The e ned mesh could be used in he compu a ion o a mo e accu a e e e ence solu ion u . Howe e , he compu a ion o he e e ence solu ion as a global e ned p oblem mus be a oided because o he la ge amoun o deg ees o eedom in ol ed. The e e ence solu ion u is associa ed wi h a e e ence e o , e := u - uh. The e e ence e o e, can be seen as he p ojec ion o he ac ual e o e on he in e pola ion space W gene a ed by he global e ined mesh. The space W includes he space V (V: = V1 EB V 2 EB· · · EB V m) because W con ains he unc ions Ni gene a ing each V k, associa ed wi h he in e io nodes, bu also he in e pola ion unc ions associa ed wi h he nodes lying on he bounda y o each elemen lk. The e o e, he p ojec ion e o e on V is also he p ojec ion o e on V and he es ima e gi en by e also unde alua es he no m o e . The goal o he p esen ed e o es ima o is o app oxima e he global and local no ms o e wi hou sol ing he global p oblem. In ac , e ( he p ojec ion o e on V) is al eady an ini ial es ima e o e ,. The global p ojec ion e is he assembly o he local p ojec ions ek on each subspace V k. Each p ojec ion ek is compu ed sol ing a linea sys em wi h a ew deg ees o eedom, hus, wi h a low compu a ional cos . Once he subspace Vi, is de ined by he basis � o in e pola ion unc ions Ni, Eqs. (7) and (8) a e used in o de o compu e lleJI. The compu a ion o he esidual e m [[], de ined in Eq. (6), is s aigh o wa d once uh belongs o he e e ence space W. Then, uh can be exp essed by he column ec o [uh]@ o i s nodal alues in (b) (a) Fig. l. Re e ence elemen a y submesh (a) and induced elemen a y submeshes o e egula (b) and a bi a y (c) meshes. 5 he elemen a y submesh disc e izing nk. Thus, a(uh, JS), see igh -hand side o Eq. (6), is easily compu ed om he p oduc o K and [uh] qe· Finally, we ge [[] = [l] -K[uh] qe (13) This exp ession de e mines he igh -hand side o Eq. (7) as a unc ion o he e ms appea ing in he usual FE app oach o a p oblem. The esidual, i.e. he igh -hand side o Eq. (7), does no need in eg a ion o he app oxima e solu ion, because in eg als ha e been ca ied ou when K was gene a ed. The esidual e m [l] associa ed wi h a subspace � wi h unc ions de ined o e an elemen nk is called in e io esidual. The es ima o gi en by Eq. (7) wi h his de ini ion o he � is called in e io es ima o . The in e io es ima o is cha ac e ized by he choice o he elemen a y submeshes. I is e y simple o de ine elemen a y submeshes o quad ila e al elemen s using a s uc u ed egula mesh o he e e ence squa e, see Fig. 1. I is impo an o no ice ha al hough he elemen a y submesh is chosen s uc u ed, he o iginal compu a ional mesh can be uns uc u ed. These kind o submeshes e i y ha : (a) hey a e simply de ined on he e e ence elemen , (b) hey can ep esen he es ic ion o uh o nk and (c) hey can be uni o mly e ined as a as he use wishes. Compu ing he p ojec ion Bk as desc ibed in Eq. (7) is equi alen o sol ing a local p oblem in nk, wi h nk disc e ized by he elemen a y submesh and whe e homogeneous Di uchle bounda y condi ions a e imposed on he bounda y o nk. The no m o Bk inc eases wi h he 'size' o he associa ed subspace� and he numbe o deg ees o eedom o he elemen a y submesh. A sequence o app oxima ions based on e ined submeshes has mono onic g ow h on no ms and i is bounded by he no m o he exac and he e e ence e o . The e o e, as he submesh is e ined, he no m o he p ojec ions (local and global) con e ges o a alue which is an unde es ima ion o he e o . This unde es ima ion could be ela i ely a om llell;, and e en om lid;, due o he ac ha we a e o cing he app oxima e unc ion o ake ze o alues a he hidden poin s. Many au ho s [11-13] sol e simila local p oblems app oxima ing he bounda y condi ions. The main idea o hose es ima o s is o use he discon inui y (jump) o lux ac oss he elemen edges o app oxima e ux (Neumann) bounda y condi ions o he e o equa ion, on he bounda y o each nk. Di e en au ho s use di e en c i e ia o deduce such app oxima ions. These app oxima ions can be expensi e because hey need o compu e he solu ion and i s de i a i es in poin s lying on he bounda y o he elemen s. F om he poin o iew o he p esen p ojec ion app oach, i is possible o ake in o accoun he in o ma ion o lux jumps ac oss edges by compu ing p ojec ions on subspaces o unc ions de ined on subdomains including he edges. In he nex sec ion we in oduce a new amily o p ojec ions based on a new pa i ion o n. Finally, we compu e a new e o es ima e combining he p ojec ions on he wo amilies. Some compa ibili y condi ions a e imposed in o de o make his combina ion possible and o s ill ge a lowe bound. 5. En ichmen o he es ima ion. O e lapped meshing Up o now, we ha e compu ed local p ojec ions Bk o he e o e on subspaces�- We ha e used a pa i ion o n in subdomains nk (k = 1, .. . , m, namely he elemen s). Each subspace Vk was de ined by a submesh o e he subdomain lk. The no ms o Bk we e aken as app oxima ions o local measu es o he ue e o e. Local and global app oxima ions, Bk and B, gi en by (8), (11) and (12), a e lowe bounds o he exac and e e ence e o no ms. We ealized ha his app oxima ion could be poo . We would like o add new e ms o he e o es ima ion (9), in o de o ge close o he exac e o no m and s ill keep he lowe bounding. Since B is he p ojec ion o he e e ence e o e, on he space V, e can be uniquely exp essed as he sum o B and a unc ion e�, o hogonal o B, such ha e, = B + e�. Using he Py hago as heo em we can w i e llell2 = lld2 -lle�l 2, he e o e he no m o e� is he unde es ima ion o he global e e ence e o by he global in e io es ima e. The goal o he en ichmen o he es ima ion is o app oxima e he o go en pa o he e e ence e o e� and o add i o he i s (in e io ) es ima es. The unc ion e� belongs o V he o hogonal space o V V : = V1 EB V2 EB · · · EB Vm, in he e e ence in e pola ion space W. As we wan o app oxima e e � by a p ojec ion, we would like o p ojec on a subspace included in V _j_. 6 Le us conside a new pa i ion o {} in o subdomains A1 ([ = 1, ... , m'). In o de o make clea he di e ence be ween he wo pa i ions, subdomains deno ed by A1 a e called pa ches, while nk a e he ac ual elemen s. E e y pa ch A1 is disc e ized by a pa ch-submesh gene a ing a subspace U1 in H�(A1). The subspaces U1 a e gene a ed in he same way as he spaces V k associa ed wi h he elemen s. In o de o app oxima e he e e ence e o , he elemen s o each pa ch-submesh a e chosen o be elemen s o elemen a y submeshes and, hen also elemen s o he e ined global mesh gene a ing W Pa ch-submeshes and elemen a y-submeshes sha e nodes and elemen s. Since subdomains A1 ha e ze o measu e in e sec ions, subspaces U1 a e mu ually o hogonal. The p ojec ions o e (o e) on he U/s can be hough o as a new amily o local app oxima ions o e o e di e en domains. Le U1 be he pa o U1 o hogonal o he in e io global es ima e E:. Since U1 is a es ic ion o U1 subjec o one linea cons ain , he dimension o he subspace U1 is, a leas , he dimension o U1 minus one. Le YJ1 deno e he p ojec ion o he e e ence e o e on U1• Since E: and each 1/i a e o hogonal Bessel's inequali y applies again, m m m llell 2 + 2: IIYJ1ll2 = I lleJ2 + I IIYJ1ll 2 � llell2 • (14) The o hogonali y o E: does no impose U1 o be included in V 1_. Ac ually, a es ic ion o U1 ensu ing o hogonali y o he space V would educe i o he null subspace. This is due o he ac ha , in gene al, he dimension o he subspace U1 is less han he o al numbe o nodes which a e in he elemen s in e sec ing he co esponding A1• Since we a e ob aining a lowe bound o he e o , we p e e o impose he minimal es ic ion allowing o add he new con ibu ions o he in e io es ima es as is done in Eq. (14). We ha e de e mined a new global es ima e. Now, he only hidden poin s a e he in e sec ions o he edges o he wo amilies o subdomains, ha is he in e sec ions o he bounda ies o he elemen s and he pa ches, which is ob iously a smalle se han all he edges o he elemen s. Ne e heless, as no ed p e iously, we also desi e o compu e a local es ima e o de e mine he spa ial a ia ion o he e o . This local es ima e is an app oxima ion o lid!, see Eq. (5). The e o e, he con ibu ion o all he pa ches A1 o e lapping elemen nk mus be aken in o accoun . This con ibu ion is no he comple e IIYJ bu he local no m es ic ed onk, namely ak(Y/1, YJ1) = IIYJill!. Recall ha 1/i has suppo on A1, he e o e i is ze o in {}k ou side i s in e sec ion wi h A1. Thus, an app oxima ion o he squa ed local no m o he e o llell! can be e alua ed using (15) whe e index I in he summa ion akes he alues such ha he pa ch A1 o e laps he elemen nk. No ice ha he local no m o 1/i ( es ic ed o A1) is easily compu ed using an exp ession simila o (8). Howe e , he compu a ion o i s con ibu ion o elemen nk, i.e. IIYJi ll k, needs o iden i y he elemen s o he pa ch-submeshes belonging o ilk. This can be cumbe some. The e o e, in o de o u he simpli y he compu a ions ano he app oxima ion is pe o med. We e e o equally dis ibu e he squa ed no m o 1/i, IIYJi ll 2 , o e all he elemen s o e lapping he pa ch A1• Tha means we a e app oxima ing he quan i ies de ined in (15) by (16) whe e m1 is he numbe o elemen s o e lapping he pa ch A1 and index l has he same ange o a ia ion as in Eq. (15). This app oxima ion looses he lowe bound p ope y, in pa icula when elemen s wi h small e o a e nex o elemen s wi h la ge e o . Howe e , he compu a ional bene i s a e impo an and he examples in he nex sec ion show ha he lowe bound p ope y is only los in a negligible numbe o elemen s. I is impo an o no ice ha he o hogonali y condi ion o he in e io global app oxima ion e can be easily implemen ed i he nodes o he pa ch-submeshes (disc e iza ion o A1) and he nodes o he elemen a y submeshes (disc e iza ion ilk) a e he same. Then, he selec ion o he subdomains A1 is induced by he disc e iza ion o nk. We mus choose a he same ime he geome y and he meshing o A1 condi ioned o he 7 Fig. 2. Node pa ch submesh (shaded) associa ed wi h he egula elemen a y submeshes. Fig. 3. Edge pa ch submesh (shaded) and elemen a y submeshes de ined h ough a quad ila e al meshing o a iangle. meshing o n. The o hogonali y condi ion becomes a linea es ic ion on he local solu ion o e each pa ch and i can be easily imposed by a Lag ange mul iplie echnique. Finally, in o de o comple ely de ine he es ima o , a desc ip ion o he new pa i ion and he co esponding pa ch-submeshes is needed. I we conside an in e io es ima o as de ined in Sec ion 4, see Fig. l, a se o pa ches cen e ed a he nodes o he o iginal mesh can be de ined as is shown in Fig. 2. E e y node has an associa ed A1 wi h e ices a he cen e s o adjacen elemen s and he middle poin s o he edges. The disc e iza ion o he pa ches is induced by he disc e iza ion o he elemen s. The hidden poin s o he o al es ima ion a e only he cen e s o he in e io edges o elemen s. The es ima o induced by his geome ic de ini ion is deno ed by nodal-pa ch es ima o . I is possible o c ea e an elemen a y-submesh and a pa ch de ini ion ensu ing ha he hidden poin s o he o al es ima ion a e a p io i chosen poin s, o ins ance he nodes o he o iginal mesh. Fig. 3 shows how such submeshes and pa ches a e de ined. I is wo h ema king ha wi h his geome ic de ini ion, p ojec ion 711 includes he e ec o lux jump ac oss he edge co e ed by he pa ch A1• This es ima o is deno ed by edge-pa ch es ima o . In he de ini ion o he es ima o we ha e ollowed a pa icula p ojec ion s a egy. In ac , we i s ob ain an in e io p ojec ion Bk o e each � and hen we use he global in e io app oxima ion B o impose a es ic ion o each U1• The sum o he p ojec ions Bk and 711 is only a lowe bound o he ull p ojec ion on a global space gene a ed by bo h he wo amilies �'s and U/s. The p ojec ion s a egy could heo e ically ail, depending bo h on he de ini ion o he submeshes and he pa ches, see [15] o de ails. Ne e heless, in he nume ical examples his pa hology has ne e been no iced. In he ollowing sec ions we show some examples and we compa e he di e ence o he nodal-pa ch es ima es and he edge-pa ch es ima es in a simple p oblem. 6. Examples The p ac ical analysis o an e o es ima o is based on he s udy o he e ec i i y index. The e ec i i y index is de ined as he a io o he measu es o he ac ual and he es ima ed e o . I can be de ined ei he globally o 8 locally. I he exac analy ical solu ion o he p oblem is known, he e ec i i y index can be compu ed. I no , in o de o s udy he pe o mance o he es ima o , he e e ence e o e can eplace he exac e o e in he compu a ion o he e ec i i y index. The lowe bound p ope y o he es ima o is equi alen o ha e e ec i i y indexes lowe han 1. The global e ec i i y index is lowe han 1 e en i i is compu ed using he e e ence solu ion ( he es ima e unde alua es he e e ence e o ). The e o e, local e ec i i y indexes a e expec ed o be also lowe han 1. We i s p esen a simple unidimensional p oblem wi h known analy ical solu ion. Le us conside he o dina y di e en ial equa ion 2 d U 2 - - (x) = 6x -3x dx2 in n = ]0, 1 [ wi h homogeneous Di ichle condi ions u(O) = u( 1) = 0. The exac solu ion o his p oblem is u(x) = (x3 -x 4) I 2. We sol e he p oblem using he FE me hod wi h egula uni o m meshes. We ob ain wo app oxima e solu ions: uh1 wi h a mesh o 20 linea ( wo-nodes) elemen s and uh2 wi h a mesh o 15 quad a ic ( h ee-nodes) elemen s. The e o s e1 = u - uh1 and e2 = u - uh2 a e known, he e o e we can compu e local ene gy no ms o he exac e o on e e y elemen o bo h meshes. In he e o es ima ion p ocedu e, we i s compu e he p ojec ion o e o on in e io subspaces gene a ed by uni o m elemen submeshes. We compu e p ojec ions o elemen a y submeshes de ined by di e en numbe s o subelemen s. Global esul s a e shown in Tables 1 and 2. Inc easing he numbe o deg ees o eedom in he elemen a y submesh he global es ima e con e ges e y quickly o he global measu e o exac e o . The dis ibu ions o local e o a e shown in Fig. 4. The es ima es p esen ed in Fig. 4 ha e been compu ed wi h coa se submeshes (5 linea elemen s and 2 quad a ic elemen s), o ine submeshes he plo s o exac and app oxima e e o dis ibu ions canno be dis inguished. The es ima ed global e o is a e y good app oxima ion o global no m o exac e o . Al hough we only ake in o accoun in e io p ojec ions and nodes a e hidden poin s, he app oxima ion is ai . This is due o he ac ha in such simple p oblems, we ha e supe con e gence phenomena. In hose p oblems FE solu ions ha e be e app oxima ion p ope ies a nodes [16]. Thus, he e o is ze o (o has a highe o de o punc ual con e gence) a nodes which a e p ecisely he hidden poin s. The in e io global app oxima ion is, in his simple case, a good one. In o he wo ds, Di ichle homogeneous condi ions on local e o equa ions a e e y accu a e. O e lapped pa ches co e ing he ex emes o elemen s would no in oduce any imp o emen in he es ima ion. The second example is a plane Laplacian p oblem wi h a simple analy ical solu ion. We conside he L-shaped domain n o Fig. 5 and we s a e he Laplacian equa ion i/u a2u -2 +-2 =4 inn. ax ay Table I Global e o es ima es and e ec i i y index o unidimensional p oblem wi h linea elemen s Elemen submesh Elemen submesh Global e o es ima e ee nodes elemen s 2 3 0.01488 4 5 0.015463 8 9 0.015687 16 17 0.015758 Table 2 Global e o es ima es and e ec i i y index o unidimensional p oblem wi h quad a ic elemen s Elemen submesh Elemen submesh Global e o es ima e ee nodes elemen s 3 2 7.3403 x 10-' 5 3 7.5345 x 10-' II 6 7.5799 x 10 3 Global e ec i i y index 94.3% 98.0% 99.4% 99.8% Global e ec i i y index 96.80% 99.40% 99.98% 9 Fig. 15. Sequence o e ined meshes leading o he op imal mesh gi ing 5% o ela i e e o . 7. Conclusions The me hodology in oduced in his pape p o ides a new app oach o e o es ima ion in FE analysis. The p esen ed lowe bound es ima o has a good pe o mance app oxima ing bo h he global alue and he shape o he e o . Al hough i is a esidual es ima o , i does no need o app oxima e bounda y condi ions o local p oblems. The e o e i is no necessa y o compu e lux jump c oss he edges o each elemen and he e o in oduced by lux spli ing echniques is a oided. One o he main disad an ages o he esidual es ima o s is hen p ecluded. Mo eo e , he es ima o is de ined elemen wise and i can be applied in meshes combining elemen s o di e en shapes. The e e ence solu ion is app oxima ed by sol ing local p oblems subjec o homogeneous Di ichle bounda y condi ions. The e o e s anda d ou ines o FE codes can be used o build up each local p oblem. Consequen ly, he es ima o is easily in eg a ed in exis ing codes. Adap i e p ocedu es using his es ima o p esen as con e gence o op imal meshes gi en a speci ied ela i e e o . Fu he s udies abou he quali y and he obus ness o he es ima o unde s ong egula i y condi ions can be pe o med ollowing he me hod desc ibed in [2] and [5]. 16 Following he same idea, di e en es ima es could be ound by de ining he local in e pola ion subspaces by highe o de polynomial unc ions. The e e ence solu ion would be hen a p- e ined one and he o hogonali y condi ions should be imposed in a di e en way. Re e ences [IJ J.Z. Zhu, E. Hin on and O.C. Zienkiewicz, Mesh en ichmen again mesh egene a ion using quad ila e al elemen s, Comm. Nume . Me hods Eng g. 9 (1993) 547 554. [2] I. Babuska, T. S ouboulis and C.S. Upadhyay, A model s udy o a pos e io i e o es ima o s o linea ellip ic p oblems. E o es ima ion in he in e io o pa chwise uni o m g ids o iangles, Compu . Me hods Appl. Mech. Eng g. 114 (1994) 307 378. [3] O.C. Zienkiewicz and J.Z. Zhu, A simple e o es ima o and adap i e p ocedu e o p ac ical enginee ing analysis, In . J. Nume . Me hods Eng g. 24 (1987) 337 357. [4] O.C. 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Diez, Un nue o es imado de! e o pa a el me odo de los elemen os ini os, Doc o al Thesis, Uni e si a Poli ecnica de Ca alunya, 1996. [16) T.J.R. Hughes, The Fini e Elemen Me hod (P en ice Hall, Englewood Cli s, NJ, 1987). [17) E. Oiia e and G. Bugeda, A s udy o mesh op imiza ion c i e ia in adap i e ini e elemen analysis, Eng g. Compu . 10 ( 1993) 307 321. [18) J. Sa a e, M. Gu ie ez and A. Hue a, A new algo i hm o uns uc u ed quad ila e al mesh gene a ion and ezoning, in: Ken Mo gan e al., eds., P oc. VIII In . Con . on Fini e Elemen s in Fluids, Ba celona (Pine idge P ess, 1993) 745 754. 17